The Kähler Kawamata–Viehweg vanishing conjecture

Let $X$ be a normal compact Kähler space of dimension $n$ with canonical singularities, and let $L$ be a nef line bundle on $X$ with numerical dimension $u(L)$.

Kähler vanishing conjecture. If $q>n- u(L)$, then

$$H^q(X,K_X+L)=0.$$

This would substantially extend the Kähler Kawamata–Viehweg-type vanishing theorem stated immediately before it, which covers the case $c_1(L)^2\ne0$ and $q\ge n-1$. The supplied text gives no resolution status.