Failure of additivity conjecture for sofic entropy

Let $\Gamma$ be a countable discrete non-amenable sofic group with sofic approximation $\Sigma$. Let $A$, $B$, and $C$ be countable $Z(\Gamma)$-modules in an exact sequence

$$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0.$$

Failure of additivity conjecture. There is such an exact sequence for which

$$h_{\Sigma}(\widehat{B},\Gamma)\ne h_{\Sigma}(\widehat{A},\Gamma)+h_{\Sigma}(\widehat{C},\Gamma).$$

This predicts that sofic entropy need not satisfy an addition formula for algebraic actions of non-amenable sofic groups; the paper presents it as an analogue of the corresponding conjectured failure for metric mean dimension.