Derived contraction algebra conjecture for stable endomorphism dgas

Let $R$ be a Gorenstein singularity, let $M$ be a maximal Cohen–Macaulay $R$-module, and set $A:=\operatorname{End}_R(R\oplus M)$. Let $e\in A$ be the idempotent corresponding to $\operatorname{id}_R$, and suppose that $A/AeA$ is an Artinian local $k$-algebra. Write $A\mathbin{/^{\mathbb{L}}}AeA$ for the derived quotient and ${\mathbb{R}}\underline{\operatorname{End}}_R(M)$ for the derived stable endomorphism dga. Derived contraction algebra conjecture. The quasi-isomorphism type of $A\mathbin{/^{\mathbb{L}}}AeA$ determines the quasi-isomorphism type of ${\mathbb{R}}\underline{\operatorname{End}}_R(M)$ as a dga over $k[\theta,\theta^{-1}]$.

The preceding proposition proves this implication when comparing two maximal Cohen–Macaulay modules whose derived quotients are quasi-isomorphic and whose ordinary quotients are Artinian local. The conjecture formulates the resulting natural expectation that the derived contraction algebra determines the periodic stable endomorphism dga; its resolution status is not supplied in the source span.