Bressoud's coefficient-positivity conjecture

For nonnegative integers $A,B$, define the $q$-binomial coefficient by

$$\begin{bmatrix} A\B\end{bmatrix}_q:= \begin{cases} \displaystyle\frac{(q;q)_A}{(q;q)_B(q;q)_{A-B}},&0\le B\le A,\\ 0,&\text{otherwise.} \end{cases}$$

Suppose that $M,N\in\mathbb Z^+$, that $\alpha$ and $\beta$ are positive rational numbers, and that $K$ is a positive integer with $\alpha K$ and $\beta K$ integers. If $1\le\alpha+\beta\le2K+1$, with strict inequalities when $K=2$, and $\beta-K\le n-M\le K-\alpha$, then the polynomial

$$\sum_{j=-\infty}^{\infty}(-1)^jq^{j(K(\alpha+\beta)j+K(\alpha-\beta))/2}\begin{bmatrix}M+N\M+Kj\end{bmatrix}_q$$

has non-negative coefficients. The source presents this as Bressoud's conjecture and does not report a resolution.