Nakamura's strong c-Wilf equivalence conjecture

Let $\mathfrak{S}_m$ denote the symmetric group on $m$ letters, and let $\pi,\tau\in\mathfrak{S}_m$. Write $\pi\sim\tau$ for ordinary Wilf-equivalence and $\pi\overset{s}{\sim}\tau$ for strong c-Wilf equivalence. Nakamura's conjecture. For all $\pi,\tau\in\mathfrak{S}_m$,

$$\pi\sim\tau\quad\text{if and only if}\quad \pi\overset{s}{\sim}\tau.$$

This conjecture asks whether ordinary Wilf-equivalence and strong c-Wilf equivalence coincide. The source notes that the analogous statement for ordinary Wilf-equivalence is already false for patterns of length $3$, and says that this conjecture appears to evade the methods developed in the paper.