Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$.

Now assume that $X$ is singular, but we consider only *locally trivial* deformations (in the sense of Flenner and Kosarew).
Then the statement about $H^1$ remains true, and so does the second one, according to Sernesi's book, Prop. 2.4.6.
However, unwinding the proof one finds that in fact a nonsingularity hypothesis is used, namely when applying the Infinitesimal Lifting Property.

Question:Is it true that $H^2(X, T_X)$ is an obstruction space for the locally trivial deformation functor $\mathrm{Def^{lt}}(X)$?

Ideally, an answer would either indicate how to fix the proof of Prop. 2.4.6 above, or give an example of a variety $X$ with $\mathrm{Def^{lt}}(X)$ singular even though $H^2(X, T_X) = 0$.