Let $X$ be a complex smooth projective variety and let $\iota: Y \hookrightarrow X$ be a smooth hyperplane section. Put $\dim(X)=n+1$. Then weak Lefschetz says that $$ \iota^\ast: H^k(X^{an}, \mathbb{Q}) \to H^k(Y^{an}, \mathbb{Q}) $$ is an isomorphism for $k \leq n-2$.

I would be interesting in the following variant:

First, instead of considering the whole $X$, I want to look at $U=X-D$, where $D$ is a simple normal crossings divisor.

Secondly, instead of taking $\mathbb{Q}$ as coefficients, I would like to look at a rank one local system $V$ of $\mathbb{C}$-vector spaces on $U^{an}$.

Assume $Y$ is a smooth section of $X$ (intersecting properly all the intersections of the irreducible components of $D$). Put $W=Y-D \cap Y$. Is it true that

$H^k(U^{an}, V) \to H^k(W^{an}, V_{|W^{an}})$

is an isomorphism for $k \leq n-2$? Same question for cohomology with compact support.