The abelian subgroup conjecture for doubly twisted mapping torus categories

Let $\mathcal{A}$ be a dg category with strictly commuting strict dg auto-equivalences $\phi$ and $\psi$, and let $M_{\phi,\psi}$ be the associated doubly twisted mapping torus category. Likewise define $M_{\phi',\psi'}$ for another commuting pair $\phi',\psi'$. Write $\operatorname{Auteq}(D^\pi(\mathcal{A}))$ for the group of auto-equivalences of $D^\pi(\mathcal{A})$. Abelian subgroup conjecture. $M_{\phi,\psi}$ is Morita equivalent to $M_{\phi',\psi'}$ if and only if the abelian subgroups $\langle\phi,\psi\rangle$ and $\langle\phi',\psi'\rangle$ of $\operatorname{Auteq}(D^\pi(\mathcal{A}))$ are the same. This is proposed as a two-parameter generalization of the preceding mapping-torus result and is motivated by the expected relation with the wrapped Fukaya category of the doubly twisted symplectic mapping torus; the supplied text gives no resolution.