The strong invertibility conjecture for L-space knots

Let $K$ be an L-space knot in $S^3$, meaning that $K$ admits a non-trivial Dehn surgery to a 3-manifold $Y$ satisfying

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

Strong invertibility conjecture. Every L-space knot in $S^3$ is strongly invertible: there is an orientation-preserving involution of $S^3$ whose fixed-point set is a circle intersecting $K$ in two points and which takes $K$ to itself.

The conjecture arose from the observation that many L-space knots are strongly invertible. It is refuted by the construction in this paper of asymmetric L-space knots in $S^3$, namely L-space knots that are not strongly invertible.