Formal dissection realization of all algebras derived equivalent to graded skew-gentle algebras

Establish that for every graded associative algebra B that is perfect-derived equivalent to a graded skew-gentle algebra A, there exists a formal dissection Δ of the graded orbifold surface associated to A such that B is isomorphic, as a graded associative algebra, to the cohomology H^•(A_Δ) of the endomorphism A∞ category A_Δ of Δ (equivalently, B arises as the endomorphism algebra of a formal generator of the partially wrapped Fukaya category of that orbifold surface).

Background

The paper develops a geometric framework in which partially wrapped Fukaya categories of orbifold surfaces model derived categories of (skew-)gentle algebras. A central theme is identifying which associative algebras appear as endomorphism algebras of formal generators obtained from suitable dissections of orbifold surfaces.

Within this framework, the authors propose that all graded associative algebras derived equivalent to graded skew-gentle algebras should be realized by formal dissections of the corresponding orbifold surfaces, i.e., via endomorphism algebras H^•(A_Δ) coming from explicit A∞ categories attached to such dissections. They note that this holds in the trivially graded gentle case and emphasize that the conjecture remains open in the graded setting.