Non-collapse of the spectral sequence for strongly invertible knots

Let $\widetilde{K}$ be a strongly invertible knot, and consider the spectral sequence of Theorem 1. To say that it collapses on the $E_k$ page means that the pages agree from that page onward.

Spectral-sequence non-collapse conjecture. There exists a strongly invertible knot $\widetilde{K}$ such that the spectral sequence of Theorem 1 collapses on the $E_k$ page for some $k\ge 3$.

The paper notes that the spectral sequence collapses on the $E_2$ page for all strongly invertible knots with at most $11$ crossings, and conjectures that this is not the general behavior. The existence of an example requiring a later page remains open in the supplied text.