Let $X$ be a projective, smooth, algebraic variety over a subfield of the complex numbers, and let $Y \hookrightarrow X$ be a smooth hyperplane section of $X$. The classical Lefschetz theorem claims that the morphism induced by restriction

$$ H^j(X) \longrightarrow H^j(Y) $$

is an isomorphism whenever $j\leq \dim X-2$. Here $H^j(\cdot)$ could be singular cohomology with rational coefficients or algebraic de Rham cohomology.

I'm interested in the following variant. Let $D$ be a normal crossings divisor on $X$ and $(E, \nabla)$ an integrable algebraic connection with regular singularities along $D$. Let $Y$ be a smooth hyperplane section of $X$ meeting properly all the intersections of the irreducible components of $D$. Denote by $H^j(X, E, \nabla)$ the hypercohomology of the de Rham complex of the connection, that is $$ H^j(X, E, \nabla):=\mathbb{H}^j(X, E \to E \otimes_{\mathcal{O}_X} \Omega^1_X(\log D) \to \cdots). $$

**Question**: Is is true that
$$
H^j(X, E, \nabla) \longrightarrow H^j(Y, E_{| Y}, \nabla_{| Y})
$$ is an isomorphism for $j \leq \dim X -2$ ?

Here

$E_{|Y}=E \otimes_{\mathcal{O}_X} \mathcal{O}_Y$

is the restriction of $E$ to $Y$ and $\nabla_{| Y}$ is the induced connection on $E_{| Y}$.

If this asking too much, are there some natural conditions on $(E, \nabla)$ for the question to have a positive answer?

Thanks for your help!