The bounding-cochain conjecture for pillowcase tangle invariants

Let $T$ be a 2-tangle, let $R_\pi(T)\looparrowright P^*$ be its holonomy-perturbed traceless character variety, and let $b\in \mathit{CF}(R_\pi(T),R_\pi(T))$. Let $R^\natural_\pi(T)$ denote the invariant for the tangle modified by the earring. Bounding-cochain conjecture. There exists an assignment associating to every 2-tangle $T$ a bounding cochain $b$ such that $(R_\pi(T),b)$ is a well-defined object and tangle invariant in the wrapped Fukaya category $\mathcal{W}(P^*)$, the assignment extends to earring-modified tangles yielding $(R^\natural_\pi(T),b)$, and for every decomposition $(S^3,\mathcal{L})=(D^3,T_1)\cup_{(S^2,4)}(D^3,T_2)$ one has

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

This proposes a wrapped-Fukaya-categorical tangle invariant whose gluing recovers reduced singular instanton homology; constructing the assignment and proving all three properties remain open.