Generalized injectivity conjecture for minimal multisegments

Let $\rho$ be a fixed cuspidal representation, let $d4c$ be an irreducible representation in $d4c\mathrm{rr}_{\rho}$, and let $\mathfrak n\in\mathrm{Mult}_{\rho}$ be minimal to $d4c$. Write

$$\mathfrak n=\left\{\Delta_1,\ldots,\Delta_r\right\}$$

in ascending order, and set $l=l_a(\Delta_1)+\cdots+l_a(\Delta_r)$. Let $D_{\mathfrak n}(d4c)$ denote the derivative associated with $\mathfrak n$, and let $d4c_{N_l}$ denote the corresponding Jacquet-module component. Generalized injectivity conjecture. The unique non-zero map

$$D_{\mathfrak n}(d4c)\boxtimes\mathrm{St}(\Delta_1)\times\cdots\times\mathrm{St}(\Delta_r)\longrightarrow d4c_{N_l}$$

is injective. This conjecturally extends the preceding proposition from a pair of linked segments to arbitrary minimal multisegments; the supplied text gives no resolution or further evidence for the claim.