The conjecture on valuations of colored p-ary partitions for exceptional exponents

Let $A_{p,k}(n)$ denote the number of $k$-colored partitions of $n$ into powers of the prime $p$, and let $u_p$ be the $p$-adic valuation. Write $\mathbb{P}_{\geq 3}$ for the primes at least $3$. Valuation conjecture. Let $p\in\mathbb{P}_{\geq 3}$, $u\in\{2,\ldots,p-1\}$ and $s\in\mathbb{N}_{+}$. Then, for $n\geq up^{s}$,

$$\nu_{p}(A_{p,(p-1)(up^s-1)}(n))\in\{1,2\}.$$

Moreover, for each $n\in\mathbb{N}_{+}$,

$$\nu_{p}(A_{p,(p-1)(up^s-1)}(pn))=\nu_{p}(A_{p,(p-1)(up^s-1)}(pn+1))=\ldots=\nu_{p}(A_{p,(p-1)(up^s-1)}(pn+p-1)).$$

This conjecture was suggested by numerical observations; the boundedness theorem preceding it establishes the corresponding exceptional family with $u=1$, while the cases with $u\geq2$ remain open.