Invariance conjecture for bordered manifolds with torus boundary

Let $M$ be a bordered 3-manifold with $n$ toroidal boundary components, equipped with a system of arcs $\mathscr{A}\subseteq M$. Let $\mathcal{X}(M,\mathscr{A})^{\mathcal{L}}$ denote the associated type-$D$ module over the relevant torus-boundary algebra. Invariance conjecture. The type-$D$ module $\mathcal{X}(M,\mathscr{A})^{\mathcal{L}}$ is an invariant of $(M,\mathscr{A})$ up to homotopy equivalence. This conjecture would establish that the module construction is an invariant of the bordered manifold together with its system of arcs. The paper notes that the modules may depend on the choice of arcs and leaves the invariance statement for future work.