The length conjecture for global integrals on $GL_n$

Let $F$ be a number field, ${\bf A}$ its ring of adeles, and let $\pi_1,\ldots,\pi_{l+2}$ be automorphic representations of $GL_n({\bf A})$, with $\pi_{l+1}$ cuspidal and $\pi_{l+2}$ represented by an Eisenstein series $E(g,s)$. Choose vectors $\varphi_i$ in the spaces of $\pi_i$, assume that the product of the central characters is trivial, and assume that none of the representations is one-dimensional. Let

$$I=\int\limits_{Z({\bf A})GL_n(F)\backslash GL_n({\bf A})}\varphi_1(g)\varphi_2(g)\ldots\varphi_{l+1}(g)E(g,s)\,dg,$$

where $Z$ is the center of $GL_n$. For each $\pi_i$, let $\mathcal O(\pi_i)$ be its associated unipotent orbit and define $\operatorname{dim}\pi_i=\frac{1}{2}\operatorname{dim}\mathcal O(\pi_i)$. The integral satisfies the dimension equation when

$$\sum_{i=1}^{l+2}\operatorname{dim}\pi_i=\operatorname{dim}GL_n-1.$$

The length conjecture. If $I$ satisfies the dimension equation and $l>1$, then $I$ is zero for every choice of data. Equivalently, a nonzero global unipotent integral satisfying the dimension equation has length $l+2=3$.

Nonzero global unipotent integrals satisfying the dimension equation are known, including the Rankin product integral; a partial classification is available, but the general assertion that the dimension equation forces $l=1$ remains open.