Vanishing conjecture for Whittaker inner integrals on $GL_n$

Let $F$ be a number field, ${\bf A}$ its ring of adeles, and let $\pi_1,\ldots,\pi_l$ be automorphic representations of $GL_n({\bf A})$, with vectors $\varphi_i$ in their spaces. Let $U_n$ be the maximal unipotent subgroup of $GL_n$, and let $\psi_U$ be a Whittaker character of $U_n(F)\backslash U_n({\bf A})$. Define

$$J=\int\limits_{U_n(F)\backslash U_n({\bf A})}\varphi_1(u)\varphi_2(u)\ldots\varphi_l(u)\psi_U(u)\,du.$$

The associated dimension equation is

$$\sum_{i=1}^{l}\operatorname{dim}\pi_i=\operatorname{dim}U_n=\frac{1}{2}n(n-1).$$

Whittaker vanishing conjecture. If $J$ satisfies this dimension equation, then $J$ is zero for every choice of data.

This is presented as the special case arising from the main length conjecture when the Eisenstein series is a minimal representation. The source does not provide a resolution of this special case.