Donaldson's long exact sequence conjecture for global monodromy

Let $(E,\pi)$ be an exact Morse fibration over $D$ with relative Maslov map $\delta_{E/D}$, let $M$ be the fibre over the base point, and let $\mu\in \operatorname{Symp}^e(M)$ be the global monodromy with canonical grading $\widetilde{\mu}$. For an admissible choice of paths with distinguished basis $\Gamma=(L_1,\ldots,L_m)$, choose gradings and let $\mathcal A=\operatorname{Lag}^{\rightarrow}(\widetilde{\Gamma})$. Donaldson's conjecture. There is a long exact sequence

$$HF^*(\widetilde{\mu},+)\longrightarrow H^*(E;\mathbb Z/2)\longrightarrow HH^*(\mathcal A,\mathcal A)\longrightarrow HF^*(\widetilde{\mu},+)$$

where the connecting map has degree one and the other maps have degree zero. The Hochschild cohomology is independent of the distinguished basis and its gradings, so the proposed sequence should give an invariant relation between global monodromy and the directed Fukaya category.