The facet-halving conjecture for asymptotic contributions of minimal transverse critical points

Let $\mathbf{w}$ be a minimal transverse critical point, and let $N(\mathbf{w})$ be its normal cone. For a direction $\mathbf{r}$, the direction may lie on a facet or in the interior of $N(\mathbf{w})$. Facet-halving conjecture. When a direction $\mathbf{r}$ is on a facet of the normal cone $N(\mathbf{w})$, then $\mathbf{w}$ contributes half as much to the asymptotics as when $\mathbf{r}$ is in the interior. This conjecture concerns the exceptional boundary cases in the asymptotic analysis of weighted reflectable walks, where the generating-function numerator vanishes at the critical point; the paper conjectures that the usual factor of one-half still applies.