The vanishing conjecture for primitive varieties

Let $V$ be an ${\cal L}$-primitive variety with $\alpha_{\cal L}(V)>0$. Let

$$\rho:X\to\overline{V}^{\cal L}$$

be a resolution of singularities such that the support of the ${\bf Q}$-Cartier divisor $\rho^*(L)^{\otimes\alpha_{\cal L}(V)}\otimes K_X$ has normal crossings.

Vanishing conjecture. For every $i>0$,

$${\rm h}^i(X,{\cal O}_X)=0.$$

In particular, ${\rm Pic}(X)$ is a finitely generated abelian group and

$${\rm Pic}(X)\otimes{\bf Q}\cong{\rm NS}(X)\otimes{\bf Q}.$$

The source says this vanishing is important for constructing Tamagawa numbers, but gives no resolution status.