Füredi's equi-sized chain partition conjecture for the Boolean lattice

Let $2^{[n]}$ denote the Boolean lattice of subsets of $[n]=\{1,\ldots,n\}$, and let a chain be a subset whose elements are pairwise comparable by inclusion. For a positive integer $n$, set

$$l=\left\lfloor\frac{2^n}{\binom{n}{\lfloor n/2\rfloor}}\right\rfloor.$$

Füredi's conjecture. For every positive integer $n$, the Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of each chain is $l$ or $l+1$.

If true, this would give a partition into chains of size $(\sqrt{\pi/2}+o(1))\sqrt n$. The conjecture remains open; a weaker result gives a partition into the same number of chains whose sizes lie between $0.8\sqrt n$ and $25\sqrt n$.