Uniform rank-symmetric chain partition conjecture for the Boolean lattice

Let $h$ be a positive integer. Let $N(h)$ be a positive integer depending on $h$, and let $n$ be an odd positive integer.

Uniform rank-symmetric chain conjecture. For every positive integer $h$ there exists a positive integer $N(h)$ such that, whenever $n>N(h)$ and $n$ is odd, the Boolean lattice $2^{[n]}$ has a partition into rank-symmetric chains in which all but at most one chain have size $2h$.

This is the rank-symmetric analogue of a result of Lonc concerning partitions into chains of almost uniform size. The source presents it as an open problem, and gives no resolution.