Merca's truncation conjecture for Jacobi's triple product

Let $R$ and $S$ be positive integers with $1\le S<R/2$, and let $k\ge 1$. Consider the theta series

$$\frac{(-1)^k}{(q^S,q^{R-S};q^R)_\infty}\sum_{j\ge k}(-1)^jq^{j(j+1)R/2-jS}(1-q^{(2j+1)S}).$$

Merca's conjecture. This theta series has non-negative coefficients.

Merca proposed this as a refinement of the averaged-truncation conjecture above. The source does not state a resolution, so its status remains open.