The question Are automorphism groups of hypersurfaces reduced ? reminded me of the following related question that I have not seen discussed.

If $X$ is a smooth projective variety over an algebraically closed field $k$ of characteristic $p$ such that the canonical bundle $K_X$ is ample, then is the automorphism group of $X$ reduced?

Some observations:

- It is equivalent to ask if $H^0(X,T_X) = 0$.
- When $X$ is a hypersurface in $\mathbb{P}^n$, this is exactly the question answered in the link above.
- If $d := \mathrm{dim}(X) < p$ and $X$ lifts flatly to $W_2(k)$, then the answer is "yes" as $H^0(X,T_X) = (H^d(X, \Omega^1_X \otimes K_X))^\vee = 0$ by Kodaira vanishing (which holds because of Deligne-Illusie-Raynaud under these hypotheses).

Arguments or references or counterexamples would be very appreciated!