Berrizbeitia–Medina–Moll–Moll–Noble conjecture on p-adic valuations of Stirling numbers

Let $k$ be a positive integer and let $p$ be a prime with $k<p$. For positive integers $m$, define

$$L_{p^m}=(p-1)p^{\lceil\log_p k\rceil+m-2}.$$

Call a congruence class $[a]_d$ constant when the relevant $p$-adic valuation of $S(n,k)$ is independent of $n\in[a]_d$, and non-constant otherwise.

Berrizbeitia–Medina–Moll–Moll–Noble conjecture. Fix a positive integer $k$ and a prime number $p$ such that $k<p$. Then there exist $m_0(k)\in\mathbb{N}_+$ and $\mu(k)\in\mathbb{N}$ such that for any integer $m\geq m_0(k)$ there are exactly $\mu(k)$ non-constant congruence classes $[a]_{L_{p^m}}$ modulo $L_{p^m}$, and each non-constant class modulo $L_{p^m}$ splits into $p-1$ constant and one non-constant class modulo $L_{p^{m+1}}$.

This conjecture generalizes the preceding $2$-adic conjecture to arbitrary primes under the hypothesis $k<p$. The supplied status evidence says that the fact was proved by Gessel and Lengyel when $k/p$ is not an odd integer; since $k<p$, that condition holds here, so the conjecture is solved.