Gluing conjecture for Fukaya-category bimodules of sutured manifolds

Let $F,F',F”$ be connected Riemann surfaces and let $Z$ be a finite subset of the identified boundaries. Let $Y_1,Y_2$ be sutured manifolds with

$$\partial Y_1=(-F)\cup F',\qquad \partial Y_2=(-F')\cup F”,$$

and let $Y=Y_1\cup_{F'}Y_2$ be obtained by gluing $Y_1$ and $Y_2$ along $F'$. Equip $F,F',F”$ with collections of disjoint properly embedded arcs $\underline{\alpha},\underline{\alpha}',\underline{\alpha}”$, and suppose that $\underline{\alpha}'$ decomposes $F'$ into discs, each containing at most one point of $Z$. Gluing conjecture. There is a quasi-isomorphism of $A_\infty$-bimodules

$$\mathcal{Y}(\mathbb{T}_{Y})\simeq \mathcal{Y}(\mathbb{T}_{Y_1})\otimes_{\mathcal{A}(\mathbb{F}',k')}\mathcal{Y}(\mathbb{T}_{Y_2}).$$

This conjecture predicts that the bimodule associated with a glued sutured manifold is obtained by tensoring the bimodules associated with its two pieces over the Fukaya-category algebra of the gluing surface. It is presented as an expected gluing property, and the supplied source gives no resolution.