Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely for any open subset $U\subset M$, $F(U)$ is the space of generalized functions on $U$.

Is it true that $H^i_Z(M,F)=0$ for any $i>0$?

Remark. I think that if one considers the sheaf of smooth functions instead of generalized, then the anologous statement is not true in general even when $M =\mathbb{R}$ and $Z$ is a point.