Hoste–Shanahan commensurability conjecture for 2-bridge knots

Let $J(m,n)$ be the 2-bridge knot in the stated family, and let $\mathbb S^3-J(m,n)$ denote its complement. A knot is fibered when its complement is a surface bundle over $\mathbb S^1$; a knot complement is commensurable to another knot complement when the two complements have a common finite-sheeted cover. A $\mathbb Z/2\mathbb Z$-homology sphere is a $3$-manifold with the same homology as $\mathbb S^3$ with coefficients in $\mathbb Z/2\mathbb Z$.

Hoste–Shanahan's commensurability conjecture. The complement $\mathbb S^3-J(m,n)$ is commensurable to a fibered knot in a $\mathbb Z/2\mathbb Z$-homology sphere if and only if $J(m,n)$ is fibered.

The conjecture extends the paper's results on twist knots and a second family of 2-bridge knots. The supplied text provides no resolution or partial proof of the general assertion.