Epsilon dichotomy conjecture for the Ginzburg–Rallis model

Let $F$ be a local field, let $\beta$ be a character of $F^{\times}$, and let $\Pi$ be an irreducible generic representation of $\mathrm{GL}_6(F)$ with central character $\beta^2$. Let $m_{GR}(\Pi)$ denote the multiplicity relative to the Ginzburg–Rallis model with character $\beta$. Epsilon dichotomy conjecture. One has

$$m_{GR}(\Pi)=1 \iff \epsilon\bigl(1/2,(\wedge^3\Pi)\otimes\beta^{-1}\bigr)=1,$$

and

$$m_{GR}(\Pi)=0 \iff \epsilon\bigl(1/2,(\wedge^3\Pi)\otimes\beta^{-1}\bigr)=-1.$$

This conjecture predicts that the Ginzburg–Rallis multiplicity is determined by the central epsilon factor of the third exterior-power representation, giving an epsilon dichotomy for generic representations of $\mathrm{GL}_6(F)$.