Multiplicity conjecture for the model $({\mathrm{GU}}_4\times {\mathrm{GU}}_2,({\mathrm{GU}}_2\times {\mathrm{GU}}_2)^0)$

Let $E=F(\sqrt{\epsilon})$ be a quadratic extension of $F$, with associated quadratic character $\eta_{E/F}$, and let $\Pi_\phi=\Pi_\phi(G)\cup(\cup_i\Pi_\phi(G_i))$ be a tempered $L$-packet with central character trivial on $Z_{G,H}(F)$. Let $\chi_\phi$ be its central character, and let $\epsilon(\frac{1}{2},\Pi_\phi,\rho_1)$ and $\epsilon(\frac{1}{2},\Pi_\phi,\rho_2)$ be the associated epsilon factors. The groups are

$$G={\mathrm{GU}}_{2,2}\times {\mathrm{GU}}_{1,1},\qquad H=({\mathrm{GU}}_{1,1}\times {\mathrm{GU}}_{1,1})^0,$$

and their pure inner forms are $(G_i,H_i)$ for $1\leq i\leq4$, with $(G_4,H_4)$ occurring only in the Archimedean case.

Multiplicity conjecture. The unique distinguished element belongs to $\Pi_\phi(G)$ if and only if

$$\chi_\phi(-1)\eta_{E/F}(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_1)=\chi_\phi(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_2)=1.$$

It belongs to $\Pi_\phi(G_1)\cup\Pi_\phi(G_4)$ if and only if

$$-\chi_\phi(-1)\eta_{E/F}(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_1)=\chi_\phi(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_2)=1.$$

It belongs to $\Pi_\phi(G_2)$ if and only if

$$\chi_\phi(-1)\eta_{E/F}(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_1)=-\chi_\phi(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_2)=1.$$

It belongs to $\Pi_\phi(G_3)$ if and only if

$$\chi_\phi(-1)\eta_{E/F}(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_1)=\chi_\phi(-1)\epsilon(\frac{1}{2},\Pi_\phi,\rho_2)=-1.$$

This predicts which pure inner form contains the unique distinguished representation in the tempered $L$-packet, refining the analogous weak conjecture for this spherical model. The supplied text does not establish the assertion or provide evidence resolving it.