Nadel vanishing for exhaustion singular-positive line bundles

Let $X$ be a weakly pseudoconvex manifold, let $L\longrightarrow X$ be a holomorphic line bundle, and let $\mathscr{J}$ be an ideal sheaf on $X$. Assume that $L$ is exhaustion singular-positive with the ideal sheaf $\mathscr{J}$. Nadel vanishing conjecture. One should have

$$H^q(X,K_X\otimes L\otimes\mathscr{J})=0$$

for every $q>0$.

This is a non-compact weakly pseudoconvex analogue of Nadel vanishing. In the setting developed in the paper, the stated vanishing can be resolved under additional hypotheses supplied by the paper's $L^2$-Dolbeault and exhaustion results.