The minus immersed-curve pairing conjecture

Let $M_1$ and $M_2$ be manifolds with torus boundary, and let $\phi:\partial M_1\to\partial M_2$ be an orientation-reversing gluing map. Let $\mathit{HF}^{-}(M_i)$ denote the decorated immersed curves associated to $M_i$, and let $\mathcal{HF}$ denote their Floer-theoretic pairing. Minus immersed-curve pairing conjecture. If $M_1$ and $M_2$ are manifolds with torus boundary and $\phi$ is an orientation-reversing gluing map, then

$$\mathit{HF}^{-}(M_1\cup_\phi M_2)\cong\mathcal{HF}\bigl(\phi(\mathit{HF}^{-}(M_1)),\mathit{HF}^{-}(M_2)\bigr).$$

This would give a general pairing theorem for the proposed minus bordered invariants. It is conditional on the immersed curves being bordered invariants and is presented as an expected future result.