Odd-prime valuation conjecture for Ā at k=p

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. Odd-prime valuation conjecture. If $p>2$ is prime and $a$ is a positive integer, then

$$\operatorname{ord}_p\left(A_{\overline{\mathcal K},p,p^a}(0)\right)=ap-2.$$

This is an empirical conjecture about the constant terms at prime-power parameters; no proof or resolution is supplied.