I would like to know if Grauer-Riemenschneider vanishing theorem is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle.

Let me recall

Grauer-Riemenschneider vanishing theroem: Let $\pi: X\to Y$ be a surjective projective morphism of two varieties with $X$ smooth. Let $K_X$ be the canonical line bundle. Then $$R^i\pi_*(K_X)= 0, \qquad \text{ if } i> \dim Y-\dim X.$$

In the book by Lazarfeld's book, positivity in algebraic geometry, this theorem follow from Kawamata-Viehweg's vanishing theorem: Let $X$ be a projective smooth variety. Then for any nef and big line bundle $L$, $$H^i(X,K_X\otimes L)=0, \qquad, \text{ for any } i>0 .$$

My question is: if we replace $K_X$ by $K_X^{\otimes n}$, $n>0$ is Grauer-Riemenschneider vanishing theorem still holds?

As pointed out by Jason Starr that Kawamata-Viehweg would fail for pluri-canonical line bundle. Still it is quite possible to some generalization, for example take $L^{\otimes r_n}$, where $r_n$ depends on $n$.