Recursive criterion for record-equivalence of permutation patterns

Fix $k\ge2$ and let $\pi,\sigma\in S_k$. For a permutation $w$, let $\mathrm{Rec}(w)$ denote its record set. Define record-equivalence by equality of the numbers of avoiders with each record set at every length. For $i\in[k]$, let $\pi^{(i)}$ and $\sigma^{(i)}$ be the standardized permutations obtained by deleting position $i$, and let $m_\pi=\pi^{-1}(k)$ and $m_\sigma=\sigma^{-1}(k)$.

Recursive criterion for record-equivalence. The permutations $\pi$ and $\sigma$ are record-equivalent if and only if all of the following hold:

1. $\mathrm{Rec}(\pi)=\mathrm{Rec}(\sigma)$.
2. $|\operatorname{Av}_n(\pi)|=|\operatorname{Av}_n(\sigma)|$ for all $n\ge k$.
3. If $m_\pi=m_\sigma$, then for every $i\in[k]$ with $i\ne m_\pi$, the permutations $\pi^{(i)}$ and $\sigma^{(i)}$ are record-equivalent in $S_{k-1}$.
4. If $k\ge4$, $\mathrm{Rec}(\pi)=\mathrm{Rec}(\sigma)=[k-1]$, and $m_\pi=m_\sigma=k-1$, then $\pi(k)=\sigma(k)$.

The criterion reduces record-equivalence to record-set agreement, Wilf-equivalence, and recursive deletion-minor conditions, with a special terminal condition. The source supplies no resolution status; it is presented as a conjecture.