Das–Saikia–Sarma 3-adic and 2-adic 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 first conjecture. For all $i,j\geq 1$, all $n\geq 0$, and positive integers $k$ not divisible by $2$ or $3$,

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

The conjecture was posed based on numerical evidence, and the source does not report a proof or disproof.