Centralizer conjecture for sofic entropy

Let $(X,\mu,T)$ be a $G$-system, and let the centralizer of $T$ be the subgroup of measure-preserving automorphisms of $(X,\mu)$ that commute with the image of the homomorphism $G\longrightarrow \operatorname{Aut}(X,\mu)$ induced by the action $T$. If the centralizer is ergodic, then

$$\mathrm{h}_\Sigma(\mu,T)=\mathrm{h}^{\mathrm{q}}_\Sigma(\mu,T).$$

If the centralizer is weakly mixing, then

$$\mathrm{h}_\Sigma(\mu,T)=\mathrm{h}^{\mathrm{dq}}_\Sigma(\mu,T).$$

Centralizer conjecture. If the centralizer of $T$ is ergodic, then the sofic entropy and quenched sofic entropy coincide; if it is weakly mixing, then the sofic entropy and doubly quenched sofic entropy coincide. The claim proposes a generalization of the role of the left-shift action in the proof of the paper's sufficient conditions; its resolution is not established in the supplied text.