Generalized Denef–Jacobs–Veys vanishing conjecture

Let $X$ be a smooth irreducible complex projective variety, let $(E_i)_{i\in S}$ be a finite set of mutually distinct prime divisors on $X$, and let $\bm a=(a_i)_{i\in S}$ with $a_i\in\mathbb Q$ satisfy

$$\sum_{i\in S}(a_i-1)E_i\sim_{\mathbb Q}K_X.$$

Set $U=X\setminus\bigcup_{i\in S}E_i$, and let $\mathcal L(\bm a)$ be the rank-one local system on $U$ whose monodromy around each $E_i$ is $e^{2\pi\sqrt{-1}a_i}$. Assume that a good log resolution of $(X,\sum_i(a_i-1)E_i)$ exists and is an isomorphism over $U$. Generalized vanishing conjecture. If

$$H^*(U,\mathcal L(\bm a))=0,$$

then

$$\operatorname{PV}(X,\bm a)=0.$$

This is proposed as a more natural version of the preceding vanishing conjecture, allowing divisors that need not have normal crossings. The source gives no resolution of the conjecture, so its status remains open.