Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and $A$, respectively. Then $a$ induces a map $f \colon \textrm{Def}(S) \to \textrm{Def}(A)$, whose differential is $f_* \colon H^1(S, T_S) \to H^1(A, T_A)$.

Since every small deformation of $S$ is again a surface of general type, if Def(S) is generically smooth then the image of $f_*$ is contained in the subspace of dimension $\frac{g(g+1)}{2}$ of $H^1(A, T_A)$ corresponding to the algebraic deformations of $A$ (here $g := \dim(A)$ ).

Is this still true when Def($S$) is not generically smooth, i.e. everywhere non-reduced? I suspect that the answer should be "yes", but I would like to see a rigorous proof (or a counterexample, if my guess is wrong).