Consider the boolean poset $2^{[n]}$. Is it true that for each rank $k< \frac{n}{2}$ and each positive integer $t\le \binom{n}{k}$, there is a perfect matching between the first $t$ elements of level $k$ of the poset when written out in lexicographical order and the first $t$ elements of level $k+1$ (also written out in that order)?
For example, for $n=5$, $k = 2$, $t=3$, we have a perfect matching between $(12, 13, 14)$ and $(123, 124, 125)$ since $125$ covers $12$, $124$ covers $14$, and $123$ covers $13$.