The Koszul-dual duality conjecture for finite-dimensional minimal $L_\infty$-algebras

Let $(\operatorname{\mathfrak g},\ell)$ be a minimal finite dimensional $L_\infty$-algebra over the field ${\mathbf{k}}$. Write $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})$ for the dg coalgebra dual to the representing pc dg algebra $\operatorname{CE}^\bullet(\operatorname{\mathfrak g})$, and let $d(\operatorname{CE}_\bullet(\operatorname{\mathfrak g}))$ denote its derived linear dual. The cocycle $\nabla(\ell)$ defines the Maurer–Cartan twist $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})^{[\nabla(\ell)]}$.

Koszul-dual duality conjecture. The derived linear dual of the dg $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})$-comodule $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})$ is weakly equivalent to the right dg $\operatorname{CE}_\bullet(\operatorname{\mathfrak g})$-comodule

$$d(\operatorname{CE}_\bullet(\operatorname{\mathfrak g}))\simeq \operatorname{CE}_\bullet(\operatorname{\mathfrak g})^{[\nabla(\ell)]}[-|\operatorname{\mathfrak g}|].$$

This is presented as the $L_\infty$-version of a previously established dual-comodule result. The conjectural extension is motivated by the Koszul-dual viewpoint, since a direct approach using universal enveloping algebras would require controlling higher $A_\infty$-terms. It remains open in the supplied source context.