Wang's 5-power congruence conjecture for the spt$_\omega$ function

Let $\textup{spt}_\omega(n)$ denote the partition statistic used in the paper. For integers $k\geq 1$ and $n\geq 0$, Wang's conjecture.

$$\textup{spt}_\omega\left(2\cdot 5^{2k-1}n+\dfrac{7\cdot 5^{2k-1}+1}{12}\right)\equiv 0\,(\textup{mod}\,5^{2k-1}),$$ $$\textup{spt}_\omega\left(2\cdot 5^{2k}n+\dfrac{11\cdot 5^{2k}+1}{12}\right)\equiv 0\,(\textup{mod}\,5^{2k}).$$

The conjecture concerns an infinite family of congruences modulo powers of $5$. The source states that it was subsequently proved using modular forms by Wang and Yang, so it is solved.