Chern's inner-series coefficient-positivity conjecture

Let $m,s\geq 1$ be integers. Consider the formal $q$-series

$$\sum_{k\geq 0}\frac{q^k(1-q^k)}{(q^s;q^m)_k}.$$

Chern's conjecture. This $q$-series has nonnegative coefficients in its expansion. The source presents this as a sufficient condition for the preceding double-series conjecture after rearranging the double sum, so it is a more general-looking auxiliary formulation of the same coefficient-positivity goal.