Ergodic product-disintegration conjecture

Let $S$ be a compact metric space. Define the left shift $T$ on $S^{\mathbb N}$ by

$$(T\boldsymbol{x})_i=x_{i+1},$$

and let $\widetilde{T}$ be the analogous left shift on $M_1(S)^{\mathbb N}$. For $\boldsymbol{\lambda}\in M_1(S)^{\mathbb N}$, write

$$\rho(\boldsymbol{\lambda})=\prod_{i\in\mathbb N}\lambda_i.$$

A measure $q\in M_1(S^{\mathbb N})$ is $T$-invariant and $T$-ergodic if and only if there exists a $\widetilde{T}$-ergodic measure $Q\in M_1(M_1(S)^{\mathbb N})$ such that

$$q=\int\rho(\boldsymbol{\lambda})\,Q(\mathrm{d}\boldsymbol{\lambda}).$$

Ergodic product-disintegration conjecture. A $T$-invariant measure on $S^{\mathbb N}$ is $T$-ergodic exactly when it admits a representation as a mixture of product measures whose mixing measure is ergodic under the induced shift $\widetilde{T}$. The paper presents this as a conjecture arising from a possible ergodic-theoretic reformulation; its resolution is not supplied here.