The p-adic valuation conjecture for differences of m-colored p-ary partition functions

Let $d_m(n)$ denote the $m$-colored $p$-ary partition function, let $p\in\mathbb{P}_{\geq3}$, and let $\nu_p$ denote the $p$-adic valuation. Write $m=pk+i$ with $k\in\mathbb{N}$ and $i\in\{0,\ldots,p-1\}$. p-adic valuation conjecture. Then

$$\nu_p(d_m(p^2n)-d_m(n))=\nu_p(n)+2\nu_p(m)+3-[p=3].$$

If $i=p-1$, then

$$\nu_p(d_{pk+p-1}(p^2n)-d_{pk+p-1}(n))\geq\nu_p(n)+4-[p=3].$$

For infinitely many values of $n$, this inequality is an equality. The conjecture predicts a sharper valuation than the congruences obtained from the preceding theorem, but no proof or resolution is supplied here.