Conjugate-slope property conjecture for L-space knots

Let $K$ be a nontrivial L-space knot in $S^3$, and let $g(K)$ be its Seifert genus. The conjugate-slope property at $n$ is the property defined in the paper for $n\in\mathbb N$. Conjugate-slope property conjecture. $K$ has the conjugate-slope property at $2g(K)-1$. The claim is motivated by the many examples established in the paper, including broad classes of positive-braid and twist-positive knots, but it remains conjectural for arbitrary nontrivial L-space knots.