Restriction conjecture for truncated-semistable pure sheaves

Let $X$ be a smooth projective variety over an algebraically closed field $k$, with a fixed very ample line bundle $\mathcal{O}_X(1)$. Let $E$ be an $\ell$-(semi)stable pure sheaf of dimension $d$ on $X$, where $\ell<d$, and let $D$ be a general divisor in $|\mathcal{O}_X(a)|$.

Restriction conjecture. The restriction $E|_D$ remains $\ell$-(semi)stable for $a\gg0$.

This would extend the paper's restriction theorems for torsion-free $\ell$-semistable and $\ell$-stable sheaves to pure sheaves supported in positive codimension. The authors note that effective restriction results are known for slope-(semi)stable pure sheaves, but the corresponding statement for general $\ell$-semistable pure sheaves remains open.