Nakamuta's conjecture on c-Wilf equivalence of permutation patterns

Let $v$ and $w$ be permutation patterns. They are c-Wilf-equivalent in permutations if

$$g_0^v({\mathcal S}_n)=g_0^w({\mathcal S}_n)$$

for every $n\in{\mathbb N}$, and they are strongly c-Wilf-equivalent in permutations if

$$g_r^v({\mathcal S}_n)=g_r^w({\mathcal S}_n)$$

for every $n\in{\mathbb N}$ and $r\in{\mathbb N}_0$, where $g_r^v({\mathcal S}_n)$ counts permutations in ${\mathcal S}_n$ containing $v$ consecutively exactly $r$ times. Nakamuta's conjecture. If two patterns are c-Wilf-equivalent in permutations, then they are also strongly c-Wilf-equivalent in permutations. This conjecture asserts that equality of consecutive-avoidance counts already determines the full distribution of consecutive occurrences for permutation patterns. It is presented as an important open problem; the analogous classification problem for classical pattern avoidance is also described as wide open.