Let $k$ be an algebraically closed field and $X$ be a normal variety. According to [Kol13], the notion of rational singularities may be defined as follows:
We say $X$ has rational singularities if for every resolution $f:X'\to X$, $R^if_*\mathcal{O}_{X'}=R^if_*\omega_{X'}=0$ for $i\ge 1$.
If $k$ is characteristic $0$, then the condition $R^if_*\omega_{X'}=0$ for $i\ge 1$ is unnecessary by the Grauert-Riemenschneider vanishing theorem. However, if $k$ is positive characteristic, Grauert-Riemenschneider vanishing theorem is false in general and it might be necessary to add such a condition in the definition of the notion of rational singularities.
My question is to define the notion of rational singularities, whether the condition $R^if_*\omega_{X'}=0$ for $i\ge 1$ is necessary or not. More concretely,
Let $X$ be a normal variety over $k$ and suppose for any resolution $f:X'\to X$, $R^if_*\mathcal{O}_{X'}=0$ for $i\ge 1$. Does this true that for any resolution $f:X'\to X$, $R^if_*\omega_{X'}=0$ for $i\ge 1$? Or equivalently, does this true that $X$ is CM? (For the equivalence, see Even Less Easy on the linked page.)
I believe that the above question turns out to be negative and tried to find a counterexample. But I can't find any counterexample. Is there a reference that proves the above question is false?
[Kol13] J. Kollár: Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, (2013).
