Conjecture on dyadic congruences for overcolored partition functions

Let $\bar{a}_{r,s}(n)$ denote the overcolored partition function with parameters $r$ and $s$. For integers $k\geq 1$ and $j,i\geq 0$, consider the parameter pair

$$r=2^{k+1}j+2^k-1,\qquad s=2^ki+1.$$

Dyadic congruence conjecture. For every integer $n\geq 0$, the congruences

$$\bar{a}_{2^{k+1}j+2^k-1,2^ki+1}(3n+2)\equiv 0 \pmod {2^{k+1}},$$ $$\bar{a}_{2^{k+1}j+2^k-1,2^ki+1}(9n+3)\equiv 0 \pmod {2^{k+2}},$$ $$\bar{a}_{2^{k+1}j+2^k-1,2^ki+1}(9n+6)\equiv 0 \pmod {2^{k+2}}.$$

These congruences extend the paper's arithmetic investigations of the overcolored partition function; the concluding remark specifically calls for an elementary proof of the congruences in this conjecture, and no resolution is supplied in the given text.