The first partial theta Borwein conjecture

Let $m$ and $k$ be non-negative integers. For $n$ a non-negative integer, write $(a;q)_n:=\prod_{j=0}^{n-1}(1-aq^j)$ and $(a_1,\dots,a_m;q)_n:=\prod_{r=1}^m(a_r;q)_n$. Define Laurent polynomials $A_{m,n,k}(q)$, $B_{m,n,k}(q)$, and $C_{m,n,k}(q)$ by

$$(q,q^2;q^3)_n\prod_{j=1}^m(p^jq,p^jq^2;q^3)_n(p^jq^{-1},p^jq^{-2};q^{-3})_n =\sum_{k\geq0}p^k\left[A_{m,n,k}(q^3)-qB_{m,n,k}(q^3)-q^2C_{m,n,k}(q^3)\right].$$

The first partial theta Borwein conjecture. For each $m,k\geq0$, there is a non-negative integer $N_{m,k}$ such that, if $n\geq N_{m,k}$, then $A_{m,n,k}(q)$, $B_{m,n,k}(q)$, and $C_{m,n,k}(q)$ have non-negative coefficients. Moreover, for $m=1$, $N_{1,k}=0$ for $k\leq4$ and $N_{1,k}=\lceil k/4\rceil$ for $k\geq5$; for $m>1$, $N_{m,k}=N_k$ is independent of $m$.

This is a partial-theta extension of Borwein's first conjecture, whose corresponding positivity statement was settled by Wang. The displayed values of $N_{m,k}$ are computationally supported, while the asserted eventual coefficient non-negativity remains conjectural.