Finiteness conjecture for bounded-genus L-space knots in twist families

Let $\{K_n\}$ be a twist family of knots, obtained by twisting along a fixed unknot disjoint from the initial knot, and let $g(K_n)$ denote the genus of $K_n$. Twist-family finiteness conjecture. For every integer $N\geq 0$, there are only finitely many L-space knots $K_n$ in the family such that $g(K_n)\leq N$. This is the restriction of the bounded-genus finiteness conjecture to twist families and is the focus of the paper. It remains open in the generality stated.