Asymptotic ratio of late-growing permutations

Prove or disprove that the ratio \(|\Phi_p|/(2p-2)!\), where \(\Phi_p\) is the set of late-growing permutations contributing to the monomial evaluation of \(\consta(p)\), converges from above to \(1\) as \(p\to\infty\).

Background

The monomial specialization reduces the computation of $\consta(p)$ to a signed sum over the contributing set $\Phi_p$ of late-growing permutations. The paper tabulates the first several values of $|\Phi_p|$ and observes that the normalized ratio $|\Phi_p|/(2p-2)!$ appears to decrease toward one.

The stated conjecture concerns the limiting behavior of the odd-indexed subsequence of late-growing permutation counts. It remains unresolved in the paper.