Varshavsky–author conjecture on the cocenter summand and Langlands-dual data

Let $Z_e^{red}$ be the quotient of $Z_e$ by its unipotent radical. Let ${\bf Z}_e\subset (Z_e^{red})^2$ be the algebraic variety of pairs of commuting elements in $Z_e^{red}$. Let ${\mathcal O}_e$ be the ring of regular conjugation-invariant functions on ${\bf Z}_e$. Let $\bar{{\mathcal O}}_e$ be the subring of functions $f$ such that, for every $x\in Z_e^{red}$, the function $f_x:y\mapsto f(x,y)$ is locally constant, descends to a function on $\pi_0(Z_e^{red}(x))$, and that descended function is a linear combination of characters of irreducible representations of $\pi_0(Z_e^{red}(x))$ occurring in the cohomology of the space ${\mathcal B}_e^x$ of simultaneous fixed points of $e$ and $x$ on ${\mathcal B}$. Varshavsky–author conjecture. There is a canonical isomorphism

$$C_e\cong \bar{{\mathcal O}}_e.$$

This conjecture gives a description of the cocenter component $C_e$, and hence of a summand of the space of invariant distributions on the $p$-adic group, in terms of the Langlands dual group. The statement is presented as a joint conjecture with Yakov Varshavsky; no resolution is supplied.