Exterior-cube epsilon-factor conjecture for the local Ginzburg–Rallis model

Let $F$ be a local field, let $\pi$ be an irreducible admissible representation of $GL_6(F)$ with central character $\chi^2$, and let $\pi_D$ be its local Jacquet–Langlands correspondence to $GL_3(D)$ when it exists, and zero otherwise. Let $m(\pi)$ denote the multiplicity of the Ginzburg–Rallis model, and let $\epsilon(1/2,\pi,\wedge^3)$ be the central value of the exterior-cube epsilon factor. Assume in addition that the central character of $\pi$ is trivial. Exterior-cube epsilon-factor conjecture.

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

and

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

This conjecture seeks to characterize the multiplicity by the sign of the central exterior-cube epsilon factor, providing an arithmetic criterion for distinction in the local Ginzburg–Rallis model. The source states it as another aspect of the local conjecture, without supplying a resolution.