BH evaporation vs CMB

This is not a duplicate, I am not asking about how hawking radiation works or how CMB works or whether BHs can really evaporate, or not. My quesion is about how CMB (considering space expansion's effect on decreasing CMB frequency) will feed infinitely BHs with photons' energy and if the Hawking radiation can win over this, if there is (at least theoretically) any calculation on this?

Hawking radiation is a black-body radiation that is predicted due to QM effects near the horizon.

Hawking radiation reduces the mass and energy of black holes and is therefore also known as black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. Micro black holes are predicted to be larger emitters of radiation than larger black holes and should shrink and dissipate faster.

Now CMB is the remnant cosmic microwave backround radiation from an earlly stage of the universe.

The CMB is faint cosmic background radiation filling all space. Precise measurements of the CMB are critical to cosmology, since any proposed model of the universe must explain this radiation. The CMB has a thermal black body spectrum at a temperature of 2.72548±0.00057 K.[4] The spectral radiance dEν/dν peaks at 160.23 GHz, in the microwave range of frequencies, corresponding to a photon energy of about 6.626 × 10−4 eV. Alternatively, if spectral radiance is defined as dEλ/dλ, then the peak wavelength is 1.063 mm (282 GHz, 1.168 x 10−3 eV photons).

What is Hawking radiation and how does it cause a black hole to evaporate?

where Andrew says:

If you set that temperature equal to the current temperature of the Cosmic Microwave Background (CMB) that is 2.725 K, then you get a mass of about 4.503 X 10^22 kg, or a little over half the mass of the Moon. Black holes above this mass will be cooler than the CMB incident upon them, so will gather mass-energy from it. Black holes below it will lose energy due to Hawking radiation faster than they gain it from the CMB, so will head towards a catastrophic, runaway "pop." Note that the CMB is also getting cooler as time goes on, so the equilibrium mass shifts upwards. No one that I know of has bothered to do any detailed "race" calculations between a black hole's Hawking radiation and the changing temperature of the CMB. Another important mass related to Hawking radiation is the mass at which the black hole is so cool that it would have emitted negligible radiation even if had been around since the beginning of the universe. This is about 2 X 10^11 kg, roughly comparable to the total mass of all humans. The second mass is less than the first, so if a whole range of black holes had been created at the beginning of the universe, the upshot is that some would be popping right now! Astronomers are on the lookout for these events.

Now what non of these are talking about is, the expansion of space.

The photons that existed at the time of photon decoupling have been propagating ever since, though growing fainter and less energetic, since the expansion of space causes their wavelength to increase over time (and wavelength is inversely proportional to energy according to Planck's relation).

https://en.wikipedia.org/wiki/Cosmic_microwave_background

Now based on this, CMB is made up of photons. Now this would mean that CMB is losing frequency, space is expanding. Space expansion is accelerating.

What I did not find anything about, is how CMB's losing energy because of expanding space is affecting how CMB can feed black holes. Is there any available at least theoretical calculation on whether BHs will evaporate because of Hawking radiation, or they will keep growing because of CMB's feeding them with photons' energy.

Question:

1. Will BHs be fed infinitely by CMB (considering space expansion's effect on decreasing CMB frequency) and win over evaporation?