Das–Saikia–Sarma second congruence conjecture for odd-part overpartition tuples

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. Das–Saikia–Sarma's second conjecture. For all $i\geq 1$, all $n\geq 0$, and integers $j$ satisfying the source's condition that $j$ is not a power of $2$, is not a multiple of $2$, and is not divisible by $3$,

$$\overline{OPT}_{3^i j}(3n+2)\equiv 0 \pmod{3^{i+1}\cdot 2}.$$

This conjecture was posed from numerical evidence, and no resolution is given in the source.