Coefficient-positivity conjecture for the refined Borwein polynomials

Let $L$ and $M$ be nonnegative integers, and let $\mathcal{B}(L,M,a,b)$ and $\mathcal{B}_{0,1}(L,M,a,b)$ denote the polynomials occurring in the displayed sums. Define

$$\mathcal{A}_{L,M}(q)=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(9j+1)/2}\mathcal{B}(L,M,3j,j),$$

and

$$\mathcal{C}_{L,M}(q)=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(9j+7)/2}\mathcal{B}_{0,1}(L,M,3j+1,j).$$

Refined Borwein coefficient-positivity conjecture. The polynomials $\mathcal{A}_{L,M}(q)$ and $\mathcal{C}_{L,M}(q)$ have nonnegative coefficients.

These conjectured finite refinements are presented as stronger-looking positivity statements, but the source notes that they do not directly imply positivity of $A_L(q)$ and $C_L(q)$. The source gives no resolution status.