A diagonal valuation formula for Stirling numbers of the first kind

Let $p$ be an odd prime. Let $a,n,k$ be positive integers with $1\le a\le p-1$ and $2\le k\le a(p-1)p^{n-1}+1$. Define $\epsilon_k=0$ if $k$ is even and $\epsilon_k=1$ if $k$ is odd, and let $\langle k\rangle$ be the integer satisfying $0\le\langle k\rangle\le p-2$ and $k\equiv\langle k\rangle\pmod{p-1}$. The diagonal valuation conjecture.

$$v_p\bigl(s(ap^n,ap^n-k)\bigr)=\begin{cases}n+(n+v_p(k))\epsilon_k-1-v_p\left(\left\lfloor\frac{k}{2}\right\rfloor\right),&k\equiv\epsilon_k\pmod{p-1},\ +(n+v_p(k))\epsilon_k+v_p\left(B_{2\left\lfloor\frac{\langle k\rangle}{2}\right\rfloor}\right),&k\not\equiv\epsilon_k\pmod{p-1}. \end{cases}$$

This is the specialization $m=n$ of the preceding conjecture, so it is merged here as a separately stated restatement only because the paper explicitly labels it as another conjecture; the source gives no resolution beyond the cases noted for the preceding conjecture.