Asymptotic equidistribution conjecture for modular Ackermann maps

For fixed $m\ge3$, consider the map

$$n\mapsto A_{2^k}(m,n)$$

from uniformly chosen $n\in\mathbb Z_{2^k}$ to $\mathbb Z_{2^k}$. Asymptotic equidistribution conjecture. As $k\to\infty$, this map approaches the uniform distribution on $\mathbb Z_{2^k}$; in particular, the probability that $A_{2^k}(m,n)$ takes any given value in $\mathbb Z_{2^k}$ converges to $2^{-k}$, up to fluctuations diminishing with $k$. The preceding heuristic and numerical observations suggest increasingly irregular and approximately uniform residue patterns for higher levels, but the precise distribution for arbitrary $k$ and $m$ remains open.