The factorization-closure conjecture for strictly singular operators on Lorentz sequence spaces

Let $d_{w,p}$ be the Lorentz sequence space. Let $\mathcal{SS}$ denote the ideal of strictly singular operators on $d_{w,p}$, let $J^j$ be the ideal generated by the formal identity $j\colon\ell_p\to d_{w,p}$, and let $J^{\ell_p}$ be the ideal of operators on $d_{w,p}$ that factor through $\ell_p$. Write overlines for norm closures. Factorization-closure conjecture. One has

$$\overline{J^j}=\mathcal{SS}\wedge\overline{J^{\ell_p}}.$$

In particular, every strictly singular operator in $L(d_{w,p})$ which factors through $\ell_p$ can be approximated by operators that factor through $j$. The surrounding section explicitly says that the authors do not know whether the relevant ideals are distinct, so this equality is presented as an unresolved question about norm-closed operator ideals.