The conjectures on 4-adic valuations of colored 4-ary 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 define

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

4-adic valuation conjectures. The following claims are conjectured: (1) If $m\equiv6\pmod 8$, then $(\varphi_4(S_{4,m}(n)))_{n\in\mathbb{N}}$ is unbounded. (2) If $m\not\equiv6\pmod 8$, then this sequence is 4-automatic. (3) With $a_n=\varphi_4(S_{4,1}(n+1))$, the listed initial values and recurrences hold. (4) With $b_n=\varphi_4(S_{4,2}(n+1))$, the listed initial values and recurrences hold. (5) For $s\in\mathbb{N}_{\geq2}$,

$$\nu_2(S_{4,2^s}(n))=s+1+(\nu_2(2n)\pmod2),$$

and the stated formula for $\varphi_4(S_{4,2^s}(n))$ follows. (6) For $s\in\mathbb{N}_{\geq3}$ and $m\in\mathbb{N}$,

$$\varphi_4(S_{4,2^sm+2^{s-1}}(n))=\varphi_4(S_{4,2^{s-1}}(n)).$$

These numerical predictions concern automaticity, unboundedness, and explicit valuation recurrences for the first composite base case; no resolution is given.