Bridge Theorem conjecture for topological and algebraic entropy

Let $S$ be a cancellative left amenable monoid acting on the compact abelian group $K$ on the left by $\gamma$. Let $\widehat\gamma$ denote the induced dual action on the Pontryagin dual of $K$. Bridge Theorem conjecture.

$$h_{top}(\gamma)=h_{alg}^r(\widehat\gamma).$$

The conjecture is proposed without the hypothesis that the abelian group dual to $K$ is torsion, extending the bridge theorem proved in the paper under that hypothesis.