Non-negativity conjecture for the $R=2$, $S=1$ truncated theta series

Let $k$ be a positive integer. Consider the formal power series

$$\frac{(-1)^{k-1}}{(q;q^{2})_\infty}\sum_{j=0}^{k-1}(-1)^jq^{j^2}(1-q^{2j+1})-(-1)^{k-1}(q;q)_\infty.$$

Non-negativity conjecture. This series has non-negative coefficients.

The series is the $R=2$, $S=1$ specialization arising in the proof of the main theorem. The source says that the cases $k=1,2,3$ follow from stronger results, while the general statement is left open.