Vanishing conjecture for affine configuration sums

Let $P^{(p,p')}$ be the parameter set used for the affine weights, and let $X_{\eta,\Lambda,\Lambda'}(q)$ be the corresponding configuration sum. Vanishing conjecture. For $\Lambda,\Lambda'\in P^{(p,p')}$,

$$\lim_{|\eta|\to\infty}X_{\eta,\Lambda,\Lambda'}(q)=0.$$

The supplied text gives this as a further conjecture following the nonnegativity claim, without stating any proof or resolution.