Stationary-measure decomposition for kinetically constrained models

Let $\mathcal{E}$ be the set of stable configurations, let $\pi$ be the Bernoulli product measure associated with the KCM, and for each $\beta\in\mathcal{E}$ let

$$\pi_{\beta}=\pi\mid\{\eta:\bp{\eta}{}=\beta\}$$

be the measure obtained by conditioning $\pi$ on the event that the associated stable configuration is $\beta$.

Stationary-measure decomposition conjecture. If $\mu$ is a stationary measure of a KCM, then there exists a probability measure $\mu_{*}$ on $\mathcal{E}$ such that

$$\mu=\int_{\mathcal{E}}d\mu_{*}(\beta)\,\pi_{\beta}.$$

The conjecture generalizes the preceding theorem for the $d$-dimensional East model, where this decomposition is proved. It proposes that every stationary KCM measure is a mixture of the equilibrium measures conditioned on stable configurations; the general KCM case remains open in the paper.