Chern's coefficient-positivity conjecture for a double $q$-series

Let $m$ be an integer with $m\geq 2$. Consider the formal $q$-series

$$\sum_{j\geq 0}\sum_{k\geq 1}\frac{q^{3j+k}(1-q^k)}{(q^m;q^m)_j(q^m;q^m)_{j+k}}.$$

Chern's conjecture. This double series has nonnegative coefficients in its expansion. This would provide a uniform $q$-series proof of the parity-bias inequalities previously established by Chern, including the exceptional case requiring a separate argument.