The maximum stable-state count conjecture for neighborhood grids

Let $P$ be a point configuration on $n^2$ points, and let $f^{(n,\ldots,n)}$ denote the number of standard Young tableaux of the square partition $(n,\ldots,n)$, equivalently the number of stable states of the identity configuration. For $n\in\mathbb{N}$, the conjecture concerns the number of stable states of arbitrary point configurations on these points. Maximum stable-state count conjecture. The number of stable states of any point configuration $P$ on $n^2$ points is less than or equal to $f^{(n,\ldots,n)}$. This would identify the identity configuration as attaining the maximum number of stable states among configurations with $n^2$ points; the source reports supporting results for $n\in\{1,2,3\}$ and computational experiments, but gives no resolution in general.