Indicator-element realization conjecture for Choquet decompositions of invariant means

Let $\Gamma$ be a discrete abelian group, let $m\in\mathcal M_{\tau}(\Gamma)$ be an invariant mean, let $\sigma$ be a Choquet decomposition of $m$, and let $E\subseteq\mathcal M_{\tau}^{\mathrm{ext}}(\Gamma)$ be $\sigma$-measurable. An indicator element of $L^2(m)$ is a function obtained as an $L^2(m)$-limit of indicator functions $1_{C_n}$ of subsets $C_n\subseteq\Gamma$. Indicator-element realization conjecture. There is a $\tau$-invariant indicator element $w\in L^2(m)$ such that

$$m(fw)=\int \nu(f)1_E(\nu)\,d\sigma(\nu)$$

for every $f\in\ell^\infty(\Gamma)$. This would realize the part of the Choquet decomposition selected by $E$ through a translation-invariant indicator element; its plausibility is motivated by the corresponding restriction construction, but the cited incompleteness phenomenon shows that an actual invariant set need not exist, and the conjecture remains open in the supplied context.