Can the question of a gravitationally accelerated charge radiation be tested experimentally? I know that the question of radiation from a gravitationally accelerated charge has been discussed extensively at Does a charged particle accelerating in a gravitational field radiate?. Yet the experimental aspect hasn't been touched up on.
My question strictly on this topic can appear dumb, yet for a layman like me it appears as a self-evident proposition: why can't it be verified by a test or some observation - whether a freely falling charge radiate photons, how strongly and relative to which frame of reference it does or does not radiate? If it cannot be tested practically, could you please explain why, if possible. Many thanks.
 A: There is some misunderstanding here.

whether a freely falling charge radiate photons, how strongly and relative to which frame of reference it does or does not radiate

if you mean a charge in free fall. 
In this calculation:, from the conclusion 

It is found that the "naive" conclusion from the principle of equivalence - that a freely falling charge does not radiate, and a charge supported at rest in a gravitational field does radiate - is a correct conclusion, and one should look for radiation whenever a relative acceleration exists between an electric charge and its electric field.  The electric field which falls freely in the gravitational  field is accelerated relative to the static charge.  The field is curved, and the work done in overcoming the stress force created in the curved  eld, is the source of the energy carried by the radiation. This work is done by the gravitational  field on the electric field, and the
  energy carried by the radiation is created in the expense of the gravitational energy of the system.

italics mine
So it is not a charge in free fall that radiates but a charge supported and stationary. To measure  experimentally on earth the electromagnetic radiation of a stationary charge is not possible 
a) because the gravitational constant is so small that any radiation will have such small energy that it cannot be detected. 
Look at the formula 


which is equivalent to the power radiated by an accelerated charged particle (Larmor formula), where the acceleration is replaced by g.

b) accumulating charges as in van der Graaff accelerators  induce a number of electromagnetic interactions which will radiate, not to ignore also the black body radiation, even in vacuum, and the electromagnetic coupling is orders of magnitude larger than the gravitational, which will inundate any signal
It is only in cosmological observations that one might need a contribution from such a mechanism , as discussed in the paper:

Motz  suggested that the huge radiation emerging from quasars may be created by charges located in the strong gravitational fields close to the surface of the
  quasars.  Although the current explanation for this phenomenon is different, radiation from charges located in strong gravitational  fields can still play a role in certain cosmological phenomena.

Edit with some more quotes for clarification:

A freely falling charge in a uniform GF follows a geodetic line in this system, and it is not subject to any external force. The electric  field of the charge follows similar geodesics. The charge and its  field both are located in the same frame of reference, and in that frame their relative situation is similar to the one existing between a static charge and its  field in a free space.  No relative acceleration exists between the charge and its electric  field, and we conclude that a freely falling charge does not radiate.

......

The electric  field of a charge supported at rest in the lab against GF seems
  static, but it is not.  The electric  field, which is an independent physical entity, is not supported with the charge, and it falls freely in the gravitational  field. There is a relative acceleration between the charge and its electric  field, the  field is curved (both in the lab system and in the freely falling system), and a stress force exists between the charge and its  field. The (freely falling) electric  field follows the system
  of reference characterized by the geodesics.

The italics (mine) explain the difference between the resting mass and the electric field. The mass reacts to the gravitational force, the field is freely falling.


The electric  field is detached from the supported charge, and it is not supported against gravity as the charge is. Hence the
  electric  field falls in a free fall, and it has an acceleration g relative to the supported charge. In the freely falling system, which also has an acceleration g relative to the supported charge, the charge is accelerated upward with an acceleration g 

They go on to demonstrate a non vanishing Poynting vector, i.e. electromagnetic radiation.
Hand waving, I see it as the mass/charge_carrying part of the particle settling to a lower gravitational level as  the energy is being radiated by the distorted electric field, and as I said it is a very small effect , significant maybe for cosmological dimensions. 
A: I think it can be tested experimentally.  Absent a gravitational field, it is possible to detect EM radiation from a moving charge: if the electrical influence of the charge at a distance is delayed relative to the position of the charge, as is expected from the finite propagation speed of disturbances in the EM field, then I'd say radiation is being detected. 
A charged sphere could be dropped on a hard plate so it bounces. The bounce would be "chirped", increasing in frequency until the bouncing stops.  The frequency would be low, perhaps as low as a few hundred hertz at its maximum.  However, that signal should be detectable with a sufficiently sensitive apparatus.
There would be three possible components of the signal: one that would occur at the moment of bounce when the sphere is accelerated, and two components corresponding to the upward motion and the downward motion, both of which would be in free fall.  
I'm not entirely sure that the two kinds of signals would be distinguishable, but considering that the frequency of the free-fall components would be somewhat independent of the details of the brief acceleration at the moment of bounce, I suspect that the components could be examined separately.
Testing whether a charged mass stationary in a gravitational field radiates is much more difficult, because there would not be a particular frequency to measure.  
A: Detecting the emitted photons from a charge accelerating in Earth's gravity looks like a hopeless task, given that the radiated power is tiny (see formula 9 in Anna's answer). But we can try to exploit the fact that the number of emitted photons always diverges no matter how small the acceleration. While very low energy photons (so-called "soft photons") cannot be detected directly, the fact that such photons have been emitted can be detected indirectly by measuring decoherence effects.
For example, one can consider doing a two slit interference experiment with electrons where one slit is put above the other slit. Electrons on paths that end on some spot on the screen will emit soft photons, paths that pass through the upper slit will do so slightly differently than paths that pass through the lower slit. The state of the electromagnetic field is similar but not exactly the same for the two paths. The overlap between the two states will depend on where on the screen the electron ends up, the squared modulus of this function multiples the interference pattern that you would get without this effect. The larger the distance between the two slits, the larger this decoherence effect due to the emission of soft photons will be.
