Prasad's conjecture on degenerate Whittaker representations of general linear groups

Let $F$ be a nonarchimedean local field with ring of integers $\mathfrak{o}$, and let $\mathfrak{o}_l=\mathfrak{o}/(varpi^l)$. Let $L$ and $E$ be unramified extensions of $F$ of degrees $n$ and $2n$, respectively, with rings of integers $\mathfrak{O}$ and $\mathcal{O}$, and write $\mathfrak{O}_l=\mathfrak{O}/(varpi^l)$ and $\mathcal{O}_l=\mathcal{O}/(varpi^l)$. The inclusions $\mathfrak{O}_l^\times\root \hookrightarrow {\rm GL}_n(\mathfrak{o}_l)$ and $\mathcal{O}_l^\times\root \hookrightarrow {\rm GL}_{2n}(\mathfrak{o}_l)$ are induced by the regular representations. For a representation $pi$ of ${\rm GL}_{2n}(\mathfrak{o}_l)$, let $pi_{N,\psi}$ denote its degenerate Whittaker space for the relevant unipotent subgroup and character. A character $\theta:\mathcal{O}_l^\times\tomathbb{C}^\times$ is strongly primitive if it has the property specified in the source, and let $pi$ be the irreducible strongly cuspidal representation associated to $\theta$. Prasad's conjecture. As a representation of ${\rm GL}_n(\mathfrak{o}_l)$, one has

$$pi_{N,\psi}\cong {\rm Ind}_{\mathfrak{O}_l^\times}^{{\rm GL}_n(\mathfrak{o}_l)}(\theta|_{\mathfrak{O}_l^\times}).$$

This conjecture extends the known $n=1$ result, where the degenerate Whittaker space is one-dimensional and identified with the central character. Strongly cuspidal representations arise from strongly primitive characters by results of Lusztig and Aubert–Onn–Prasad, but the asserted description for general $n$ is the conjectural part.