Lonc–Elzobi's nice-property conjecture for products of chains

For positive integers $n_1,\ldots,n_r$, let $\mathbf{n_i}$ denote a chain of length $n_i$, and let a lattice be nice when every stable-partition type dominates only types that are also realized by stable partitions. Lonc–Elzobi's conjecture. For any positive integers $n_1,n_2,\ldots,n_r$, the product $\mathbf{n_1}\times\mathbf{n_2}\times\cdots\times\mathbf{n_r}$ is nice. The source notes that Lonc and Elzobi proved the assertion for products of two chains, while the case of more than two chains is presented as an open problem.