Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference
in the literature…

Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector bundles with trivial determinant over a curve $C$ , so $X$ is not compact !) and
let $L$ be a line bundle over $X$. Then we have the standard exact sequence

$$0 \to O_X \to Diff^1(L) \to T_X \to 0$$

now take the long exact sequence of cohomology

$$... \to H^1(O_X) \to H^1(Diff^1(L)) \to H^1(T_X) \to H^2(O_X) \to ...$$

Since $Pic(X) = \mathbb{Z}$ , we obtain $H^1(O_X) = 0$. Do we also have $H^2(O_X) = 0$, so that the
middle arrow is an isomorphism ?
It is natural to expect that

$$dim (H^1(T_X)) = dim (H^1(C, T_C)) = 3g-3,$$

so that any infinitesimal deformation of the moduli
comes from a deformation of the curve, but I can find the result only for the coprime case (Narasimhan, Ramanan) and not trivial determinant.

Is this true for trivial determinant as well?

Answer

Attribution
Source : Link , Question Author : IMeasy , Answer Author : Community