Finiteness conjecture for dihedral knot surgeries

Let $m$ and $n$ be positive integers, and consider the Seifert-fibered dihedral manifolds

$$\left(-1; \frac{1}{2}, \frac{1}{2}, \frac{m}{n}\right).$$

A knot surgery is a Dehn surgery on a knot in the $3$-sphere, and the family is considered with fixed finite first homology order $|H_1(Y)|$. Finiteness conjecture for dihedral knot surgeries. Each family of dihedral manifolds with a fixed $|H_1(Y)|$ includes finitely many knot surgeries. In particular, if $n>2m+1$, then

$$\left(-1; \frac{1}{2}, \frac{1}{2}, \frac{m}{n}\right)$$

is never a knot surgery. The conjecture is motivated by the observation that all known examples of these manifolds that are knot surgeries satisfy $n\leq 2m+1$, with the cases $n=2m\pm1$ realized by torus knots. Its general status is not resolved in the supplied text.