Functoriality of the rigid refined local Langlands correspondence

Let $G_1\xrightarrow{\eta}G_2$ be a morphism of quasi-split connected reductive groups with central kernel and abelian cokernel, let $(\mathcal T,\bar h)$ be a $Z$-rigid inner twist for $G_1$ inducing $(\mathcal T',\bar h')$ for $G_2$, and let $\mathfrak w$ be the induced Whittaker datum. Assume that both groups have connected centers. For a tempered parameter $\varphi$ for $G_2$, let $\rho\in\operatorname{Irr}(\pi_0(S_\varphi^+),\mathcal T')$ and suppose that $\iota_{\mathfrak w,G_2}^{-1}(\varphi,\rho)\circ\eta'$ is irreducible.

Functoriality conjecture. The representation

$$((\mathcal T,\bar h),\iota_{\mathfrak w,G_2}^{-1}(\varphi,\rho)\circ\eta')$$

lies in the compound $L$-packet $\Pi^{Z'}_{{}^{L}\eta\circ\varphi}$. Moreover, if

$$\pi_0(S_\varphi^+)\xrightarrow{{}^{L}\eta}\pi_0(S_{{}^{L}\eta\circ\varphi}^+)$$

is bijective, then

$$((\mathcal T,\bar h),\iota_{\mathfrak w,G_2}^{-1}(\varphi,\rho)\circ\eta')=\iota_{\mathfrak w,G_1}^{-1}({}^{L}\eta\circ\varphi,\rho\circ{}^{L}\eta).$$

This is a functoriality requirement for the rigid refined local Langlands correspondence under morphisms with central kernel and abelian cokernel. The paper notes that it is weaker than a cited broader functoriality conjecture, which is known in many cases, but retains the assumptions needed for the construction.