Four remaining 2-adic congruences for overpartition tuples with odd parts

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. The four-congruence conjecture. For all $i\geq 1$, $n\geq 0$, and odd $r>0$,

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

These are the four remaining congruences from a previously proposed set of seven; the source states that they have not been proved, so their general validity remains open.