Conjecture on the unramified Langlands correspondence and elliptic pairings

Let $G$ be a reductive group, let ${\mathcal C}(G^\vee)_{\mathsf{ss}}$ be the set of semisimple conjugacy classes in the dual group, and for $s\in{\mathcal C}(G^\vee)_{\mathsf{ss}}$ and unipotent $u\in{\mathcal C}({\mathcal G}_s)_{\mathrm{un}}$ let $A_{{\mathcal G}_s}(u)$ be the relevant component group. Let $\overline R(A_{{\mathcal G}_s}(u))$ denote its elliptic quotient, and let $\overline R_{\mathrm{un}}(G')$ denote the elliptic quotient of the unipotent representation ring of an inner twist $G'$. Unramified Langlands correspondence conjecture. The unramified Langlands correspondence induces isometric isomorphisms

$$\overline{\mathsf{LLC}}_{{\mathrm{un}}}\colon \bigoplus_{s\in {\mathcal C}(G^\vee)_{\mathsf{ss}}}\bigoplus_{u\in {\mathcal C}({\mathcal G}_s)_{{\mathrm{un}}}}\overline R(A_{{\mathcal G}_s}(u))\longrightarrow \bigoplus_{G'\in{\mathrm{InnT}}(G)}\overline R_{{\mathrm{un}}}(G'),$$

and

$$\overline{\mathsf{LLC}^p}_{{\mathrm{un}}}\colon \bigoplus_{s\in {\mathcal C}(G^\vee)_{\mathsf{ss}}}\bigoplus_{u\in {\mathcal C}({\mathcal G}_s^p)_{{\mathrm{un}}}}\overline R(A_{{\mathcal G}_s^p}(u))\longrightarrow \bigoplus_{G'\in{\mathrm{Inn}}^p(G)}\overline R_{{\mathrm{un}}}(G'),$$

where the left-hand spaces have the elliptic inner products $(~,~)^{\delta_u^s}_{\mathrm{ell}}$ and the right-hand spaces have the Euler–Poincaré pairings ${\mathrm{EP}}_{G'}$. This conjecture predicts compatibility between the elliptic representation-theoretic and Langlands-parameter descriptions, but the supplied text gives no resolution evidence.