Vanishing theorems in positive characteristic

Question

In the paper

Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078

I found the following

Theorem. Let $X$ be a smooth and projective scheme over a field $k$ with $\textrm{char}(k) = p > 0.$ Assume that $X$ admits a lift to $W_{2}(k)$. If $L$ is an ample line bundle on $X$ then $$H^{i}(X,L^{-1}) = 0 \: \: \textrm{for} \: \: i < \min(p, \, \dim X).$$

I just wanted to ask if anyone knows a theorem like this under weaker hyphoteses. More precisely I have the following

Question. Does such result hold for $X$ normal and $L$ big and nef?

Answer 1

In the book Lectures on Vanishing Theorems by Esnault and Viehweg (Birkhäuser 1992) this is stated as an open problem. See in particular Problem 11.7 page 132.

However, they are able to prove the result when $\dim X=2$, since in this case one can perform the embedded resolution of singularities for curves on surfaces. See in particular Proposition 11.5 p. 129 and the subsequent remarks.