Equivariant log-concavity conjecture for rook placement loci

For $0\le r\le\min\{m,n\}$, let ${\mathcal{Z}}_{n,m,r}$ be the locus of $n\times m$ rook placements with exactly $r$ rooks, and let $R({\mathcal{Z}}_{n,m,r})$ denote its orbit harmonics ring. The left and right actions of ${\mathfrak{S}}_n$ and ${\mathfrak{S}}_m$ make this a graded ${\mathfrak{S}}_n\times{\mathfrak{S}}_m$-module. Equivariant log-concavity conjecture. For every $0\le r\le\min\{m,n\}$, the orbit harmonics ring

$$R({\mathcal{Z}}_{n,m,r})$$

is ${\mathfrak{S}}_n\times{\mathfrak{S}}_m$-log-concave. Since ${\mathcal{Z}}_{n,n,n}={\mathfrak{S}}_n$, this extends Rhoades's conjecture. It has been verified computationally for $n\le8$ and $m\le10$, but remains unproved in general.