The universal abelian cover criterion for hyperbolic knot surgeries

Let $K$ be a hyperbolic knot in $S^3$, and let $X(K)(n\mu+q\lambda)$ denote the Dehn surgery with coprime integers $n,q$ satisfying $nq\ne0$. Its universal abelian cover is the regular cover corresponding to the abelianisation homomorphism of its fundamental group. If $K$ is fibred, let $h$ denote its monodromy and $c(h)$ its fractional Dehn twist coefficient.

Universal abelian cover conjecture. If the universal abelian cover of $X(K)(n\mu+q\lambda)$ is not excellent, then $K$ is fibred and

$$q\in\begin{cases} \{nc(h)\},&\text{if }nc(h)\in\mathbb Z,\\ \{\lfloor nc(h)\rfloor,\lfloor nc(h)\rfloor+1\},&\text{if }nc(h)\notin\mathbb Z. \end{cases}$$

This predicts that failure of excellence occurs only at the exceptional surgery coefficients associated with fibred knots and their monodromies. The surrounding results establish the corresponding sufficiency statement for hyperbolic fibred knots, while the converse remains the proposed generalisation to all hyperbolic knots.