Jiang's exterior-cube epsilon-factor conjecture for the Ginzburg–Rallis model

Let $F$ be a local field and let $\pi$ be an irreducible admissible representation of $\operatorname{GL}_6(F)$ satisfying the assumptions of Jiang's multiplicity-one conjecture, with trivial central character. Let $m(\pi)$ denote the dimension of the Ginzburg–Rallis model Hom space, and let $\epsilon(1/2,\pi,\wedge^3)$ be the central value of the exterior-cube epsilon factor. Jiang's conjecture. One has

$$m(\pi)=1\iff\epsilon(1/2,\pi,\wedge^3)=1,$$

and

$$m(\pi)=0\iff\epsilon(1/2,\pi,\wedge^3)=-1.$$

This conjecture seeks to characterize the Ginzburg–Rallis multiplicity by the exterior-cube root number. The source says that it was expected from the work of Ginzburg and Rallis and first mentioned by Jiang; its resolution is not indicated in the supplied text.