Finiteness conjecture for knots sharing the Floer homology of an L-space knot

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. Let $\widehat{HFK}(K)$ denote its hat knot Floer homology. L-space-knot finiteness conjecture. There are only finitely many other knots whose Floer homology is isomorphic to $\widehat{HFK}(K)$. If true, this would give finiteness results for knots yielding a fixed lens space by Dehn surgery and would place L-space knots between positive-braid and strongly quasipositive-braid classes.