Modular-three congruences for overcubic partitions

Let $\bar a_m(n)$ denote the overcubic partition function indexed by $m$, as in the paper. For every $n\geq 0$ and $i\geq 1$, the following congruences are conjectured:

Overcubic partition congruence conjecture modulo powers of three.

$$\bar a_{3i+2}(9n+2)\equiv 0 \pmod{6},\qquad \bar a_{9i+5}(9n+3)\equiv 0 \pmod{12},$$ $$\bar a_{9i+8}(9n+3)\equiv 0 \pmod{12},\qquad \bar a_{3i+2}(9n+5)\equiv 0 \pmod{6},$$ $$\bar a_{3i+2}(9n+8)\equiv 0 \pmod{6}.$$

The conjecture concerns congruences modulo a prime other than the powers of $2$ studied in detail elsewhere in the paper, and the source suggests that its proof should be similar to the proof of the cited theorem.