Saikia's parity and 2-adic congruence conjecture for overpartition prime-tuples

Let $\overline{p}_{q}(n)$ denote the number of overpartition $q$-tuples of $n$, where $q$ is prime. Saikia's conjecture. For all $n\geq 0$ and primes $q$,

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

These congruences extend known arithmetic properties of overpartition tuples and were proposed by Saikia; their general validity remains open.