Prasad's adjoint L-function criterion for degenerate Whittaker vanishing

Let $F$ be a non-Archimedean local field, let $G=\operatorname{GL}_{2n}(F)$, and let $P_{n,n}=MN$ be the standard parabolic subgroup with Levi factor $M\simeq\operatorname{GL}_n(F)\times\operatorname{GL}_n(F)$. For a fixed nondegenerate character $\psi$ of $N$, write $\pi_{N,\psi}$ for the twisted Jacquet module of an irreducible smooth representation $\pi$ of $G$. Define

$$\operatorname{pord}_{s=t}L(s,\pi\times\pi^\vee)$$

to be the order of the pole at $s=t$ of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. Prasad's conjecture. The twisted Jacquet module vanishes if and only if

$$\pi_{N,\psi}=0$$

if and only if

$$\operatorname{pord}_{s=t}L(s,\pi\times\pi^\vee)\geq n-t+1\qquad(1\leq t\leq n).$$

The paper proves that vanishing of $\pi_{N,\psi}$ implies the predicted pole inequalities, but gives an explicit counterexample to the converse for $\operatorname{GL}_4(F)$; hence the conjecture is refuted.