On log-concavity of the number of orbits in commuting tuples of permutations

Denote by $A(p,n,k)$ the number of commuting $p$-tuples of permutations on $[n]$ that have exactly $k$ distinct orbits. It was conjectured in~\cite{abdesselam2023log} that $A(p,n,k)$ is log-concave with respect to $k$ for every $p\ge 2,n\ge 3$, and the log-concavity was proved in ``$p=\mathrm{\infty }$" case. In this paper, we prove that for $k=n-\alpha$, the log-concavity for $A(p,n,k)$ holds for every $p\ge 2$ for sufficiently large $n$.