Uniqueness of toroidal generators up to shift

Prove that within any toroidal subcategory F(X)_{λ,Q} of the monotone Fukaya category—defined as a summand split-generated by a monotone Lagrangian torus equipped with an isolated critical point of its superpotential—any two toroidal generators are quasi-isomorphic as objects of F(X)_{λ,Q}, up to a shift.

Background

The paper defines a toroidal subcategory as a summand of the monotone Fukaya category split-generated by a monotone Lagrangian torus equipped with an isolated critical point of its superpotential, with such generators called toroidal generators (Definition 1.7). Using matrix factorization techniques (Theorem C) and an automatic generation result (Theorem D), the author shows that any non-zero object split-generates a toroidal subcategory.

Despite this generation property, the fine classification of generators remains open: the conjecture asks whether any two toroidal generators for the same toroidal subcategory are actually equivalent up to a shift. The paper proves a partial result (Theorem E) showing a stable isomorphism after taking direct sums with 2^{n-1} copies in degrees 0 and 1, but the exact quasi-isomorphism up to shift is not established.