Factorization conjecture for derivative embeddings

Let $\pi\in\mathrm{Irr}_{\rho}$, let $\mathfrak n\in\mathrm{Mult}_{\rho}$ be minimal to $\pi$, and let $\mathfrak h=\mathfrak{hd}(\pi)$. Set $l_1=l_{\mathrm{abs}}(\mathfrak n)$ and $l_2=l_{\mathrm{abs}}(\mathfrak{hd}(\pi))-l_{\mathrm{abs}}(\mathfrak n)$. Let $D_{\mathfrak h}$ and $D_{\mathfrak n}$ denote the corresponding derivatives, let $\widetilde{\lambda}$ denote co-standard representations, let $\mathfrak r(\mathfrak n,\pi)$ be the removal multisegment, and let $N_{a,b}$ and $N_{a,b,c}$ denote the unipotent radicals of the indicated parabolics. Assume the minimal-model conjecture and the multisegment embedding conjecture hold, so that the maps $\iota_1$, $\iota_2$, and $\iota_3$ in the displayed diagram are defined by the stated embeddings.

Factorization conjecture. The composite $\iota_2\circ\iota_1$ factors through $\iota_3$.

This conjecture predicts compatibility between the minimal-model and multisegment embedding constructions. It is conditional on the two conjectures named in the source, whose status is not resolved in the supplied text.