How did we know that relativistic momentum is conserved in the first place? From what I understand, people historically did a lot of experiments (or just life experience) realizing that a fast moving object of small mass has the same quantity of "something" that a slow moving object of large mass also has.
Eventually this "something" came to be called momentum and it was eventually defined as: 
$$ p = mv$$ 
The fact it is conserved, is something that can be experimentally verified (to a degree) say with pool balls on a marble table. 
When studying special relativity its easy to see that traditional Newtonian momentum doesn't behave as the usual conserved quantity since mass is an unbounded positive real variable but velocity is not, so once an object has a fixed mass it has an upper bound on its possible Newtonian momentum.
There are some good intuitive reasons for wanting to re-define the velocity of an object by it's "proper velocity", (it can now be unbounded and other inertial observers, regardless of their time dilation will agree on what the relative proper velocity between two points is).
And so its natural to consider the following expression as a candidate for a conserved quantity, which we shall call relativistic momentum.
$$ p =  \frac{mv}{\sqrt{1- \frac{v^2}{c^2}}} $$
My question:
How were we experimentally or mathematically confident that this quantity is conserved? I assume no one was playing with relativistic pool balls and relativistically colliding masses, so there's no way to experimentally verify this, at least back when it was first stated.
Yet it seems like most of the physics community was confident it was conserved. Why were they so confident, given it could not be experimentally verified?
Is there some theoretical "proof of conservation" that I do not understand?
 A: 
How were we experimentally or mathematically confident that this quantity is conserved?

It is difficult to know why another person’s attitude about something is formed, however, we can look and see what possible strong justifications could have been given at the time. 
I think that the strongest theoretical justification is from Noether’s theorem. In relativity you can form a quantity called the four momentum. The symmetry of the laws of physics under translations in both space and time leads, through Noether’s theorem, to the conservation of the four-momentum. In other words, if relativity holds and if the laws of physics are the same yesterday and today and here and there then the relativistic momentum is conserved. 
Arguments by symmetry are generally considered quite strong, so this would have been seen as quite convincing. 
A: In transition-energy nuclear or particle physics it is common to use the elastic processes like 
$$e + p \to e + p$$ to establish detector performance baselines for use in more elaborate reactions like 
$$e + A \to e + p + B^* \;, $$
 where $A$ is a non-trivial target nucleus and nuclear $B^*$ is the remnant after proton knock-out.
The relationship between the scattering angles and energies of the final-state particles in the elastic baseline is entirely determined by conservation rules, and it is tested to high precision on a regular basis.
