Strong Wilf-equivalence is characterized by trivial equivalence for binary words

Let $p$ and $q$ be binary words. They are strongly Wilf-equivalent if $B_{n,p}(k)=B_{n,q}(k)$ for all $n,k\geq 0$, and they are trivially equivalent when they are related by the trivial equivalence defined earlier in the paper.

Strong Wilf-equivalence conjecture. Two binary words $p$ and $q$ are strongly Wilf-equivalent if and only if they are trivially equivalent.

Strong Wilf-equivalence refines ordinary Wilf-equivalence, since words of the same length are Wilf-equivalent. The paper notes that strong Wilf-equivalence forces $p$ and $q$ to have the same length and the same number of runs of each size, while computations suggest that trivial equivalence gives all such equivalences.