Quadratic-threshold conjecture for poset partitions of the Boolean lattice

Let $P$ be a finite poset with a unique minimum and maximum, and suppose that $|P|=2^k$ for some integer $k$. Let $c$ be an absolute constant.

Quadratic-threshold conjecture. There exists a constant $c$ such that, whenever

$$n\geq c|P|^2,$$

$2^{[n]}$ has a $P$-partition.

The paper proves the same conclusion under the weaker quantitative bound $n\geq c|P|^8$ for an absolute constant, while the quadratic dependence is known to be optimal up to a constant factor when $P$ is a chain. The conjecture remains open for general posets.