Generalized truncated Jacobi triple product conjecture

Let $R$ and $S$ be positive integers with $1\leq S<R$, let $k>\ell\geq1$, and assume $k\geq4$. There exists an integer $N$ such that the theta series

$$\frac{(-1)^{k-1}\left(\sum_{j=-k+1}^{k}-\sum_{j=-\ell+1}^{\ell}\right)(-1)^j q^{Rj(j+1)/2+Sj}}{(q^S,q^{R-S},q^R;q^R)_\infty}$$

has nonnegative coefficients for $n\geq N$. Generalized truncated Jacobi triple product conjecture. Under these conditions, the asserted eventual coefficientwise nonnegativity holds. This is proposed as a generalized truncated version of the Andrews–Merca–Guo–Zeng conjecture. The paper also gives an equivalent formulation in terms of the partition function $P_{R,S}(n)$, but does not prove the conjecture.