I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow.

Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion for some $n>0$, $U \subset X$ an open subset and $Z \subset X$ a local complete intersection subscheme. Denote by $j:U \to \mathbb{P}^n$ the natural immersion. Let $\mathcal{F}$ be a locally free sheaf on $X$. Is $H^k_{Z \cap U}(j_*(\mathcal{F}|_U)) \cong H^k_{Z \cap U}(\mathcal{F}|_U)$ for all $k \ge 0$?

N.B. If necessary, one can assume that $Z$ is smooth.