The instanton immersed-curve invariants conjecture for pointed 2-tangles

Let $T$ be a pointed 2-tangle and let $P^*\cong\partial(D^3,T)$ be the pillowcase. Write $I(T)$, $I^\natural(T)$ and $I^\sharp(T)$ for the proposed instanton-theoretic immersed curve invariants. Instanton immersed-curve invariants conjecture. There exists an assignment

$$(D^3,T)\longmapsto I(T),I^\natural(T),I^\sharp(T)\looparrowright P^*\cong\partial(D^3,T)$$

which, for every decomposition $(S^3,\mathcal{L})=(D^3,T)\cup_{(S^2,4)}(D^3,T')$, satisfies

$$\widehat{I}(\mathcal{L})=\mathit{HF}(I(T),I(T')),$$ $$I^\natural(\mathcal{L})=\mathit{HF}(I(T),I^\natural(T')),$$ $$I^\sharp(\mathcal{L})=\mathit{HF}(I(T),I^\sharp(T')).$$

These conjectural curves would recover the three indicated instanton homologies by wrapped Lagrangian Floer homology, paralleling existing Heegaard Floer and Khovanov curve invariants. Their construction and gluing formulas remain open.