Conjectured congruences for overpartition tuples with odd parts

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples with odd parts of size $n$. Let $i\geq 1$ and let $r$ be odd, with $n\geq 0$ and $n,i\in\mathbb{Z}$. Conjectured congruences. One has

$$\begin{aligned} \overline{OPT}_{2^ir}(8n+1)&\equiv 0\pmod{2^{i+1}},\\ \overline{OPT}_{2^ir}(8n+2)&\equiv 0\pmod{2^{2i+1}},\\ \overline{OPT}_{2^ir}(8n+3)&\equiv 0\pmod{2^{i+3}},\\ \overline{OPT}_{2^ir}(8n+4)&\equiv 0\pmod{2^{2i+4}},\\ \overline{OPT}_{2^ir}(8n+5)&\equiv 0\pmod{2^{i+2}},\\ \overline{OPT}_{2^ir}(8n+6)&\equiv 0\pmod{2^{2i+3}},\\ \overline{OPT}_{2^ir}(8n+7)&\equiv 0\pmod{2^{i+4}}. \end{aligned}$$

These congruences are proposed as a counterpart to the paper's theorem for overpartition tuples with odd parts, based on numerical evidence; their general validity remains open.