Baruah–Begum conjecture on congruences for the smallest parts function of restricted overpartitions

Let $\overline{\operatorname{spt}}_\omega(n)$ denote the smallest-parts function of restricted overpartitions. For integers $\ell\geq1$, $k\geq0$, and $n\geq0$, consider its values at the indicated arithmetic progressions. Baruah–Begum's conjecture. For any $\ell\geq1$, $k\geq0$ and $n\geq0$,

$$\overline{\operatorname{spt}}_\omega\left(5^{2k+\ell-1}(10n+5)\right)\equiv \overline{\operatorname{spt}}_\omega\left(5^{\ell-1}(10n+5)\right)\pmod{5^\ell},$$

and

$$\overline{\operatorname{spt}}_\omega\left(5^{2\ell}(10n+3)\right)\equiv\overline{\operatorname{spt}}_\omega\left(5^{2\ell}(10n+7)\right)\equiv0\pmod{5^{2\ell}}.$$

These conjectured congruence families extend congruences previously proved by Baruah and Begum modulo small powers of $5$ for the restricted-overpartition smallest-parts function.