Covering and stabilisation conjecture for Fukaya categories

Let $M$ be a Liouville manifold, let $\widetilde M\to M$ be a cover, and let $W:M\times D^2\to D^2$ be given by $W(m,z)=z^2$. Let $\mathcal F$ denote the relevant Fukaya categories, with vertical functors induced by inverse images of Lagrangians. Covering and stabilisation conjecture. For each cover $\widetilde M$ of $M$, there should be a commutative diagram

$$\begin{array}{ccc} \mathcal F(\widetilde M)&\longrightarrow&\mathcal F(\widetilde M\times D^2,W)\\ \uparrow&&\uparrow\\ \mathcal F(M)&\longrightarrow&\mathcal F(M\times D^2,W) \end{array}$$

where the horizontal arrows are equivalences and the vertical arrows are pullback functors assigning to a Lagrangian its inverse image. Fukaya categories of these stabilisations are not fully developed in the source's discussion; the conjecture is used to formulate consequences for geometric automorphisms after stabilisation.