3-adic valuation conjecture when m is divisible by 3

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term, and let $d_3(k)$ be the sum of the base-3 digits of $k$. 3-adic valuation conjecture. If $m\equiv0\pmod3$, then

$$\operatorname{ord}_3\left(A_{\overline{\mathcal K},k,m}(0)\right)=k\operatorname{ord}_3(m)+d_3(k)-k.$$

This is stated among the source's empirical conjectures for other values of $m$ and remains unresolved in the supplied text.