Uniqueness conjecture for the removal embedding

Let $\rho$ be the fixed cuspidal representation and let $\mathrm{Mult}_{\rho}$ denote the corresponding multisegments. For $\mathfrak h\in\mathrm{Mult}_{\rho}$, let $\widetilde{\lambda}(\mathfrak h)$ be its co-standard representation, and let $\mathfrak n$ be minimal to $\mathfrak h$. Write $\mathfrak r(\mathfrak n,\mathfrak h)$ for the removal multisegment, $l_{\mathrm{abs}}(\mathfrak n)$ for its absolute length, and $N_l$ for the relevant unipotent radical.

Unique embedding conjecture. The embedding

$$\widetilde{\lambda}(\mathfrak r(\mathfrak n,\mathfrak h))\boxtimes\widetilde{\lambda}(\mathfrak n)\hookrightarrow\widetilde{\lambda}(\mathfrak h)_{N_l},\qquad l=l_{\mathrm{abs}}(\mathfrak n),$$

exists and is unique.

The conjecture strengthens the removal-embedding proposition by asserting uniqueness, not merely existence. The paper notes that this does not follow formally from multiplicity one for standard representations.