Congruence conjecture for overpartitions into nonmultiples of 4 and 9

Let $\overline{R_{4,9}}(m)$ denote the number of overpartitions of $m$ into parts divisible by neither $4$ nor $9$. Congruence conjecture. For all $n\geq 0$, $\ell\geq 2$, and $1\leq k\leq \ell$, we have

$$\overline{R_{4,9}}(4\ell n+4k)\equiv 0 \pmod 6.$$

This conjecture proposes a further congruence for the overpartition function studied in the paper; its status is not resolved in the supplied text.