The Montesinos rational unknotting conjecture

Let $K=\mathcal{M}(e; p_1/q_1,p_2/q_2,p_3/q_3)$ be a Montesinos knot, and let $u_q(K)$ denote its proper rational unknotting number. For the three conditions below, write $\Sigma(K)$ for the double branched cover of $S^3$ over $K$:

1. $u_q(K)=1$.
2. $\Sigma(K)$ arises by $p/q$-surgery on a torus knot for some $p/q\in\mathbb{Q}$ with $q$ even.
3. $K$ can be written as $\mathcal{M}(0;a/b,c/d,r/s)$, where

$$\frac{b}{a}+\frac{d}{c}=\pm\frac{1}{ac}$$

and $s$ is even.

Montesinos rational unknotting conjecture. These three conditions are equivalent for all Montesinos knots.

The theorem preceding this conjecture proves the equivalence for three-strand Montesinos knots satisfying $|p_1|,|p_2|,|p_3|>1$ and $u(K)\geq 5$. The conjecture extends that characterization to all Montesinos knots; the paper explains that it would follow from Gordon's conjecture on non-integer Seifert fibered surgeries on hyperbolic knots.