Induction conjecture for Floer cohomology of eventually Lagrangian submanifolds

Let $c,c'$ be matching paths, let $\Lambda_c$ and $\Lambda_{c'}$ be the associated eventually Lagrangian submanifolds, and let $\mathcal A,\mathcal A'$ be directed $A_\infty$-categories with corresponding spherical objects $C\in D^b(\mathcal A)$ and $C'\in D^b(\mathcal A')$. Choose an equivalence $G:D^b(\mathcal A)\to D^b(\mathcal A')$ obtained from the chain of category equivalences described in the source. Induction conjecture.

$$HF(\Lambda_c,\Lambda_{c'})\cong\bigoplus_{d\in\mathbb Z}\operatorname{Hom}^d_{D^b(\mathcal A')}(G(C),C').$$

It is also part of the conjecture that the right-hand side is independent of the choices of admissible paths, Hurwitz moves, and quasi-isomorphisms used to construct $G$. The statement is intended to relate geometric Floer cohomology to morphisms between spherical objects in derived Fukaya categories.