The generalized partial theta Borwein conjecture

Let $m_1,m_2,n_1,n_2,n_3,$ and $k$ be non-negative integers. Define Laurent polynomials $A(q)=A_{m_1,m_2,n_1,n_2,n_3,k}(q)$, $B(q)=B_{m_1,m_2,n_1,n_2,n_3,k}(q)$, and $C(q)=C_{m_1,m_2,n_1,n_2,n_3,k}(q)$ by

$$(q,q^2;q^3)_{n_1}\prod_{j=1}^{m_1}(p^jq,p^jq^2;q^3)_{n_2}\prod_{j=1}^{m_2}(p^jq^{-1},p^jq^{-2};q^{-3})_{n_3} =\sum_{k\geq0}p^k\left[A(q^3)-qB(q^3)-q^2C(q^3)\right].$$

The generalized partial theta Borwein conjecture. For fixed $k$, if $m_1,m_2\geq1$ and $n_1,n_2,n_3$ are sufficiently large, then $A(q)$, $B(q)$, and $C(q)$ have non-negative coefficients.

This conjecture extends the first partial-theta Borwein conjecture by allowing the positive and negative $q$-shifted-factorial truncations to have separate parameters. The source gives no proof or resolution, and its formulation contains a duplicated “if,” so the intended quantifier structure should be checked against the published version.