The parabolic Kazhdan–Lusztig conjecture for Hecke algebras with separated multicharge

Let $e=\infty$ or, more generally, $e>n$, and let $\kappa\in I^\ell$ have no repeated entries. Let $\Bbbk$ be a field of characteristic $p\gg\ell$. For $\lambda,\mu\in\mathscr{P}^{\ell}_{n}$, write $n_{\lambda\mu}(t)$ for the associated parabolic Kazhdan–Lusztig polynomial of type $A_{n-1}\times A_{n-1}\times\dots\times A_{n-1}\subseteq A_{\ell n-1}$. Parabolic Kazhdan–Lusztig conjecture. The decomposition numbers of $A(n,\theta,\kappa)$ are

$$d_{\lambda\mu}(t)=n_{\lambda\mu}(t).$$

This is a stronger conjecture for the $e=\infty$ case, where the first $ep$-alcove restriction disappears and no characteristic bound depending on the number of columns is required. Its general positive-characteristic validity remains open.