Atiyah–Floer type conjectures for cyclic Cerf decompositions

Let $[Y]=[Y_{01}]\circ\cdots\circ[Y_{(k-1)k}]$ be a Cerf decomposition of a closed $3$-manifold $Y$ into simple cobordisms $Y_{i(i+1)}$, and let $L_{Y_{i(i+1)}}$ denote the associated Lagrangian correspondences. Atiyah–Floer type conjecture for cyclic Cerf decompositions. There should be isomorphisms

$$HF(L_{Y_{01}},\ldots,L_{Y_{(k-1)k}})\simeq HF_{\rm inst}(Y),$$

and

$$HF^{\cdots}(L_{Y_{01}},\ldots,L_{Y_{(k-1)k}})\simeq HF^{\cdots}_{\rm mon}(Y).$$

This extends the Heegaard-splitting version to Cerf decompositions and is intended to make decomposition-independence manifest through the symplectic $2$-categorical formalism. The source presents the extension as not yet established in the relevant generality.