Unbounded common surgery slopes among pairs of knots

Let $N$ be a natural number. Let $K(n)$ denote the $n$-surgery on a knot $K$.

Unbounded-slope conjecture. For any natural number $N$ there exist pairwise-different integers $n_1,\ldots,n_N$ and non-isotopic knots $K$ and $K'$ such that, for $i=1,\ldots,N$, $K(n_i)$ is orientation-preserving diffeomorphic to $K'(n_i)$.

This is an equivalent quantified formulation of the claim that there is no bound on the number of slopes for which some pair of knots can have common surgeries. The paper provides examples with four common surgeries, but the arbitrary-$N$ statement remains open.