Conjecture on generic distinguished representations of general linear groups

Let $F$ be the underlying field, and let $\tau$ be an irreducible generic representation of $\mathrm{GL}_n(F)$. A representation is distinguished when it has the distinguishedness property considered in the paper. Let $\Delta_1,\ldots,\Delta_m$ be representations of $\mathrm{GL}_{n_1}(F),\ldots,\mathrm{GL}_{n_m}(F)$, respectively, and write $\Delta_i^{\vee}$ for the contragredient representation. Then $\tau$ is isomorphic to the representation parabolically induced from $\Delta_1\otimes\cdots\otimes\Delta_m$. Generic distinguished representation conjecture. The representation $\tau$ is distinguished if and only if it is isomorphic to a representation parabolically induced from $\Delta_1\otimes\cdots\otimes\Delta_m$ such that each $\Delta_i$ is essentially square integrable, there is an integer $0\leq k\leq\lfloor n/2\rfloor$ with

$$\Delta_{2i}=\Delta_{2i-1}^{\vee}\quad\text{for }1\leq i\leq k,$$

and $\Delta_i$ is distinguished for $2k+1\leq i\leq n$. The conjecture would give a combinatorial characterization of irreducible generic distinguished representations, extending the preceding results on principal series and upper heredity; its resolution is not specified in the source.