The immediate-successor conjecture for the ideal generated by the formal identity

Let $d_{w,p}$ be the Lorentz sequence space, and let $cmathcal K(d_{w,p})$ denote the ideal of compact operators on it. Let $j\colon\ell_p\to d_{w,p}$ be the formal identity, and write $J^j$ for the operator ideal generated by $j$ and $\overline{J^j}$ for its norm closure. Immediate-successor conjecture. The ideal $\overline{J^j}$ is the only immediate successor of $\mathcal K(d_{w,p})$. The paper presents this as an open question concerning the lattice of norm-closed operator ideals; no resolution is supplied in the given text.