A conjecture of Nadji, Ahmia and Ramírez on congruences for biregular overpartitions

Let ${\bar{B}}_{s,t}(n)$ denote the number of overpartitions of $n$ where no part is divisible by $s$ or $t$, with $s$ and $t$ being coprime. By establishing the exact generating functions of a family of arithmetic progressions in ${\bar{B}}_{4,3}(n)$, we prove that for any $k\ge 1$ and $n\ge 1$, \begin{align*} \overline{B}_{4,3}\big(2^{k+3}n\big)\equiv0\pmod{2^{3k+5}}. \end{align*} This significantly generalizes a conjectural congruence family posed by Nadji, Ahmia and Ramírez (Ramanujan J. 67 (1):13, 2025) recently. Moreover, we conjecture that there is an infinite family of linear congruence relations modulo high powers of $2$ satisfied by ${\bar{B}}_{4,3}(n)$.