Rank-symmetric nearly equal chain decomposition conjecture for the Boolean lattice

Let $n$ be a positive integer. A rank-symmetric chain in the Boolean lattice $2^{[n]}$ is a chain whose numbers of elements in complementary ranks are equal.

Rank-symmetric chain conjecture. The Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains such that the size of any two chains differ by at most $2$.

This is proposed as a rank-symmetric variant of Füredi's conjecture. The source gives no resolution, so the conjecture remains open.