Alanazi–Munagi–Saikia's congruence conjecture for $(4,9)$-regular overpartitions

Let $\overline{R_{4,9}}(n)$ denote the number of $(4,9)$-regular overpartitions of $n$. For integers $n\geq0$, $\ell\geq2$, and $1\leq k\leq\ell$, Alanazi–Munagi–Saikia's conjecture.

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

The conjecture predicts infinite families of congruences modulo $6$ for $(4,9)$-regular overpartitions. The paper proves the stronger congruence $\overline{R_{4,9}}(4n)\equiv0\pmod{12}$ for $n\geq1$, thereby confirming this conjecture.