Jiang's multiplicity dichotomy conjecture for the local Ginzburg–Rallis model

Let $F$ be a local field, let $D$ be the quaternion algebra over $F$, and let $m(\rho)$ and $m(\rho_D)$ denote the multiplicities for the Ginzburg–Rallis models of $GL_6(F)$ and $GL_3(D)$, respectively. For an irreducible admissible representation $\rho$ of $GL_6(F)$, let $\rho_D$ be its local Jacquet–Langlands correspondence to $GL_3(D)$ when it exists, and let $\rho_D=0$ otherwise; in particular, $\rho_D=0$ if $F={\mathbb C}$. Assume that the central character of $\rho$ is $\chi^2$. Jiang's conjecture. For every irreducible generic representation $\rho$ of $GL_6(F)$,

$$m(\rho)+m(\rho_D)=1.$$

The conjecture predicts that exactly one of the split and inner-form Ginzburg–Rallis models has multiplicity one. Multiplicity-one results are known for both models, but the general relation under local Jacquet–Langlands correspondence is presented in the source as a conjecture; it can also be formulated using Vogan packets and pure inner forms of $PGL_6$.