The conjectures on valuations for non-prime-power colored partition functions

Let $S_{k,m}(n)$ be the $n$th coefficient in the power-series expansion defining the $m$-colored $k$-ary partition function, and let

$$\varphi_k(n)=\max\{s\in\mathbb{N}:k^s\mid n\},\qquad \varphi_k(0)=+\infty.$$

Non-prime-power valuation conjectures. If $k$ is not a power of a prime number, then for every $m\in\mathbb{N}_{+}$, the sequence $(\varphi_k(S_{k,m}(n)))_{n\in\mathbb{N}}$ is unbounded and is not $k$-regular. These claims are based on numerical computations and remain open.