Kawamata–Viehweg-type vanishing for exhaustion singular-positive line bundles

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

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

hold for every $q>0$? This is a proposed extension of vanishing results to weakly pseudoconvex manifolds under exhaustion singular-positivity; the asserted higher-cohomology vanishing is not established in the supplied text.