The situation is similar to this question
Let $D$ be a reduced projective scheme over $\mathbb{C}$ whose associated analytic space $D_{an}$ is simply connected as a topological space i.e. $\pi_1 (D_{an}) = 1$. Let $\Omega^1_D$ be its K\"{a}hler differential sheaf and $\hat{\Omega}_D^1$ be its double dual.
Question Is $H^0(D, \hat{\Omega}_D^1) = 0$? Is there some counterexample?
I'm considering how to deduce the above statement from the fact $\mathbb{H}^0 (D, \underline{\Omega}_D^1) = 0$ where $\underline{\Omega}_D^1$ is the graded piece of the Du Bois complex $\underline{\Omega}_D^{\bullet}$. But I don't know the way up to now.