Nonnegative-coefficient conjecture for affine configuration sums

Let $P^{(p,p')}$ be the parameter set used for the affine weights, let $P_{+}^{r}$ denote the dominant weights at level $r$, and let $X_{\eta,\Lambda,\Lambda'}(q)$ be the configuration sum defined in the paper. Nonnegativity conjecture. If

$$p'-p-n+1\geq0,$$

and $\Lambda,\Lambda'\in P^{(p,p')}$ satisfy

$$\Lambda'+(p-1)(p'/p-1)\Lambda_0\in P_{+}^{p'-p-n+1},$$

then $X_{\eta,\Lambda,\Lambda'}(q)$ is a polynomial with nonnegative coefficients. The claim is motivated by extensive computer-assisted experiments, and no proof or resolution is supplied here.