Would A Biological "Cold Haber" Process Deplete an H2/N2 World of its H2 or N2?

Based on my reading on some science papers, large, wet terrestrial planets likely exist that are swathed in a thin atmosphere of nitrogen/hydrogen ($H_2$ & $N_2$). If life evolved on such worlds, they may very well develop a biological version of the "Haber Process" that is used to industrially manufacture ammonia ($NH_3$). Since the biological version of the process would need to run at much cooler temperatures than the kiln-hot industrial Haber Process, the biological version can be called a "Cold Haber" process.

It would look like this:

$${3H_2 + N_2 \rightarrow 2NH_3}$$

At least that's the contention of some papers I've read (see references below).

Question in Full:

On an ${N_2}$/${H_2}$ atmosphere world would a Cold Haber process utilized by organisms run until it completely sequestered the non-dominant major gas (either ${N_2}$ or ${H_2}$) as ammonia? Or would something intervene to set some type of equilibrium long before the atmosphere was appreciably depleted of either ${N_2}$ or ${H_2}$?

Basically...what might be the equilibrium atmosphere and why?

You may postulate the evolution of a biological process to utilize the ammonia, but need not. If you think the evolution of organisms using a complementary process is likely or feasible and want to include that by all means include it. That certainly would effect the answer to the question!

The answer is the difference between an atmosphere of ≈99% Hydrogen/Nitrogen with traces of ammonia in both air and water, and oceans saturated with ammonia with an atmosphere chock full of it.

If needed here are parameters to run with:

PRE-COLD-HABER ATMOSPHERE

• ${H_2}$ & ${N_2}$ (90%+ of atmosphere in any ratio of 10:1 hydrogen:nitrogen all the way to 4:1 nitrogen to hydrogen)
• ${H_2O}$ Vapor (≈1%)
• ${CH_4}$ (0.01 - 5%)
• Other trace to minimally-present compounds may include ${CO_2}$, ${Ar}$, etc.

PLANET

• Oceans, continents, and some volcanic activity, much like earth
• Less UV received than Earth (1/3 at most, probably much less)
• Significant magnetic field
• Temperature: (I'd like the answer to account for more than a single planet's likely temperature range, but if needed let's go with a mean temperature between ${-40°C}$ and ${20°C}$ (your choice).
• Atmosphere between ${1bar}$ and ${20bar}$ (your choice)

OTHER BIOLOGICAL PROCESSES

• Such a world may evolve methanogenesis, converting plentiful hydrogen and outgassed $CO_2$ to Methane and Water (${4H_2 + CO_2 \rightarrow CH_4 + 2H_2O + 193 kJ}$ per mol at ${25°C}$), deriving easy energy. This would likely mean the atmosphere's supply of carbon dioxide would almost entirely convert to methane.
• Such a world may evolve photosynthesis utilizing the following chemical reaction: ${CH_4 + H_2O + y \rightarrow CH_2O + 2H_2}$, converting methane and hydrogen in the atmosphere to biomass and water.

In your answer please explain in detail your thought process. It should contain a discussion of the relevant chemical processes as well as what you suspect the new equilibrium atmosphere may be. If you can supply equations or calculations to back your answer, all the better!

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References:

Photosynthesis in Hydrogen-Dominated Atmospheres "“ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4284464/

BIOSIGNATURE GASES IN H2-DOMINATED ATMOSPHERES ON ROCKY EXOPLANETS "“ https://iopscience.iop.org/article/10.1088/0004-637X/777/2/95/meta

A BIOMASS-BASED MODEL TO ESTIMATE THE PLAUSIBILITY OF EXOPLANET BIOSIGNATURE GASES "“ https://iopscience.iop.org/article/10.1088/0004-637X/775/2/104#apj480437s4