Stopped wrapped Fukaya category and a deformed preprojective algebra

Let $X_{\mathrm{res}}$ be the resolution of the $A_n$ surface singularity, let $D_r$ be the indicated divisor, and let $\sigma$ be a stop. Let $s$ be a formal parameter, with coefficients in $\mathbb{C}((s))$, and define

$$\lambda_i(s)=q_1(s)\cdots q_{i-1}(s)(q_i(s)-1),$$

where $q_i(s)=e^{s\lambda_i}\in\mathbb{C}((s))$ and the nonzero $\lambda_i$ are determined by the symplectic areas of the non-exact Lagrangian spheres. Stopped Fukaya equivalence conjecture.

$$D^{\pi}\mathcal{W}(X_{\mathrm{res}}\smallsetminus D_r,\sigma)\cong D^b(\Pi^{\lambda(s)}(\widetilde{A}_n)).$$

Thus the category should be equivalent to the bounded derived category of finitely generated modules over the deformed preprojective algebra. This is the conjectural non-exact analogue of the exact computation via a Chekanov--Eliashberg algebra; the required wrapped Fukaya-category foundations and computation are not established in the paper.