The polyhedrality conjecture for the effective cone of primitive varieties

Let $V$ be an ${\cal L}$-primitive variety with $\alpha_{\cal L}(V)>0$. Let $\Lambda_{\rm eff}(V,{\cal L})$ denote its ${\cal L}$-cone of effective divisors in ${\rm Pic}(V,{\cal L})\otimes{\bf R}$.

Polyhedrality conjecture. The cone $\Lambda_{\rm eff}(V,{\cal L})$ is a rational finitely generated polyhedral cone.

The source describes this as a weaker statement implying the vanishing-related claim in dimensions at most $3$, together with the canonical ${\bf Q}$-Fano contraction conjecture. The supplied text gives no general resolution status.