If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about $H^j(i^!\mathcal F)$ for $j > \operatorname{codim}_X Z$? Specifically, if $\mathcal F$ is constructible on $\mathbb R^n$ and $i$ is the embedding $\mathbb R^{n-1} \hookrightarrow \mathbb R^n$, does $H^j(i^! \mathcal F)$ vanish for $j > 1$?

(The analoguous statement for coherent sheaves is: if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.)