A degenerate Whittaker criterion for $\mathrm GL_{2n}$

Let $F$ be a non-Archimedean local field. Let $N$ be the unipotent radical of the standard parabolic subgroup of $\mathrm GL_{2n}(F)$ of type $(n,n)$ with fixed nondegenerate additive character $\psi$. For an irreducible admissible representation $\pi$ of $\mathrm GL_{2n}(F)$, a theorem due to Gomez--Gourevitch--Sahi on generalized Whittaker models gives a criterion for the vanishing of the twisted Jacquet module $\pi_{N,\psi}$ in terms of the wave-front set. We translate this orbit-theoretic answer into Langlands--Zelevinsky data: if $\pi=L(\mathfrak m)$, then $\pi_{N,\psi}=0$ if and only if the Zelevinsky dual $\mathfrak m^{\mathrm t}$ contains a segment of length at least $n+1$. We do this in response to a conjecture proposed by D.Prasad about the vanishing of $\pi_{N,\psi}$ in terms of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. We prove that, for every irreducible representation $\pi$, vanishing of $\pi_{N,\psi}$ implies the pole inequalities predicted by D.Prasad. However, we show that the converse implication is false by an explicit counterexample for $\mathrm GL_4(F)$. For the generalized Steinberg constituents $v_{P_\beta}^G$ of the principal series containing the trivial representation, we make an explicit calculation of when $\pi_{N,\psi}$ is zero. In particular, for $\mathrm GL_6(F)$, exactly three of the $32$ constituents of such a principal series violate the converse direction of the conjecture proposed by D.Prasad.