Cazassus–Herald–Kirk–Kotelskiy conjecture on bounding cochains

Let $T$ be a 2-tangle, let $R_\pi(T)$ be its holonomy-perturbed traceless character variety, and let $P^*$ be the pillowcase with its singular points removed. A bounding cochain is an element $b\in CF(R_\pi(T),R_\pi(T))$ satisfying the Maurer–Cartan equation. Cazassus–Herald–Kirk–Kotelskiy conjecture. There is an assignment of a bounding cochain $b$ to every such $T$ and $R_\pi(T)\looparrowright P^*$ such that $(R_\pi(T),b)$ is a well-defined tangle invariant in the wrapped Fukaya category of $P^*$; the assignment extends to earring-modified tangles, producing $(R^\natural_\pi(T),b)$; and, for every decomposition

$$(S^3,L)=(D^3,T_1)\cup_{(S^2,4)}(D^3,T_2),$$

the resulting Lagrangian Floer homology satisfies

$$HF((R_\pi(T_1),b_1),(R^\natural_\pi(T_2),b_2))\cong I^\natural(L).$$

This conjecture proposes bounding-cochain data as the missing structure needed for a tangle invariant whose Floer homology recovers reduced singular instanton homology; the source does not state a resolution.