The minus bordered-curve complement invariance conjecture

Let $K\subset Y$ be a knot with complement $M=Y\setminus\nu(K)$. Let $\mathit{HF}^{-}(Y,K)$ denote the decorated curves constructed from minus knot Floer homology in the marked torus $T_M$. Minus complement invariance conjecture. For any knot $K\subset Y$ with complement $M=Y\setminus\nu(K)$, the decorated curves $\mathit{HF}^{-}(Y,K)$ in $T_M$ are an invariant of $M$. If true, this would provide a geometric interpretation of minus bordered Floer invariants for manifolds with torus boundary; the paper gives evidence from the integer-surgery formula, while the general conjecture remains open.