The Khovanov-theoretic characterization conjecture for L-space knots

Recall that an L-space is a rational homology sphere $Y$ satisfying

$$\dim\widehat{\operatorname{HF}}(Y)=|H_1(Y;\mathbb{Z})|,$$

and a knot in $S^3$ admitting an L-space surgery is called an L-space knot. A strong inversion on a knot is an involution of the knot pair with the standard strong-inversion symmetry, and $\varkappa(K,h)$ is the associated invariant, graded by $\delta=u-q$. L-space knot characterization conjecture. A non-trivial knot $K$ in $S^3$ is an L-space knot if and only if it admits a strong inversion $h$ and $\varkappa(K,h)$ is supported in a single diagonal grading $\delta=u-q$. This proposes a Khovanov-theoretic characterization of L-space knots, complementing the open problem of topologically characterizing L-spaces and L-space knots; it remains open.