The almost-uniform extremal conjecture for Sperner partition systems

Let $\mathcal{P}^n_k$ be the family of $k$-partitions of an $n$-set, and call a partition system Sperner when the collection of all its classes is a Sperner set system. An almost-uniform partition has class sizes differing by at most one.

Almost-uniform extremal conjecture. For positive integers $n,k$, a largest Sperner partition system in $\mathcal{P}^n_k$ is an almost-uniform partition system.

The paper proves an upper bound using the LYM inequality but leaves the exact cardinality and structure of extremal systems open.