Congruence conjecture for prime-colored overpartitions

Let $\overline{p}_{-q}(n)$ denote the number of $q$-colored overpartitions of $n$, where $q$ is prime. Congruence conjecture. For all $n\geq 0$ and primes $q$, one has

$$\begin{aligned} \overline{p}_{-q}(8n+1) &\equiv 0 \pmod{2},\\ \overline{p}_{-q}(8n+2) &\equiv 0 \pmod{4},\\ \overline{p}_{-q}(8n+3) &\equiv 0 \pmod{8},\\ \overline{p}_{-q}(8n+4) &\equiv 0 \pmod{2},\\ \overline{p}_{-q}(8n+5) &\equiv 0 \pmod{8},\\ \overline{p}_{-q}(8n+6) &\equiv 0 \pmod{8},\\ \overline{p}_{-q}(8n+7) &\equiv 0 \pmod{32}. \end{aligned}$$

The conjecture generalizes the proved congruences for $q=5,7,11,$ and $13$ established earlier in the paper, and predicts further congruences for all prime numbers of colors. Its general case remains open in the supplied source.