Nath–Saikia–Sarma's quadratic congruence conjecture for $2$-regular $3$-tuple partitions

Let $T_{\ell,k}(n)$ denote the number of $k$-tuple partitions of $n$ whose components are $\ell$-regular partitions, and write $T_2(n)=T_{2,3}(n)$. Let $p\geq 5$ be a prime satisfying

$$\left(\dfrac{-2}{p}\right)_{L}=-1.$$

Let $t$ be a positive integer such that $(t,6)=1$ and $p\mid t$. Nath–Saikia–Sarma's conjecture. For all $n\geq 0$ and $1\leq j\leq p-1$,

$$T_2\left(9\cdot t^2n+\frac{9\cdot t^2j}{p}+\frac{57\cdot t^2-1}{8}\right)\equiv 0\pmod{6}.$$

This conjecture extends the authors' preceding congruence for primes $p\equiv5$ or $7\pmod 8$ and $p\nmid n$. Its resolution is not indicated in the source.