Merca's positivity conjecture for truncated Jacobi theta series

Let $R$ and $S$ be integers with $1\le S<R/2$, and let $k\ge1$. Consider the theta series

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

Merca's conjecture. This theta series has non-negative coefficients. The conjecture refines the established positivity theorem for averaged truncations of Jacobi's triple product identity; its resolution is not specified in the supplied text.