Canonical splitting and unramifiedness compatibility conjecture

Let $\mathcal{K}$ be the set of relevant subgroups, let $H_\varphi$ be the image defined in the preceding diagrams, and let \phi_\overline{\overline{\overline{\overline{\chi}}}} be the parameter of an unramified genuine character. Let \mathcal{S}(\phi_\overline{\overline{\overline{\overline{\chi}}}}) be its component group. Unramified packet compatibility conjecture. There is a canonical splitting $s_\varphi$ of $\widehat{\Gamma_{G_{Q,n}}^{\rm tor}}$ over $\widehat{\Gamma_{G_{Q,n}}^{\rm tor}}/H_\varphi$, together with a bijection

\mathcal{L}(\phi_\overline{\overline{\overline{\overline{\chi}}}})\longleftrightarrow\operatorname{Irr}(\mathcal{S}(\phi_\overline{\overline{\overline{\overline{\chi}}}})),

written \pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\rho)\leftrightarrow\rho. For every $K\in\mathcal{K}$, splitting $s_K:K\hookrightarrow\overline{G}$ agreeing with $s_{Q,n}$ on $\mathbf{T}(O)$, $f_z\in\operatorname{Hom}(K,\mu_n)^\natural$, and $y\in\widehat{\Gamma_G^{\rm tor}}$ such that $y\cdot s_K$ is associated with $s_K$, one has that \pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\rho) is $(K,s_K)$-unramified if and only if \pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\gamma_{z,y}\cdot\rho) is $(y\cdot K,(y\cdot f_z)\otimes(y\cdot s_K))$-unramified, where

$$\gamma_{z,y}=\bigl(\varphi\circ h\circ i_K^\natural(f_z)\bigr)\cdot\bigl(s_\varphi\circ\varphi(y)\bigr).$$

This would describe how unramifiedness transforms under the dual torus and character actions.