Lonc's conjecture on almost tiling the Boolean lattice with copies of a poset

Let $P$ be a poset. The Boolean lattice $2^{[n]}$ consists of all subsets of $[n]=\{1,\ldots,n\}$ ordered by inclusion. A copy of $P$ is an induced subposet isomorphic to $P$, and a partition into copies of $P$ is a collection of pairwise disjoint copies whose union is the specified set.

Lonc's conjecture. If $n$ is sufficiently large and $|P|$ divides $2^n-2$, then

$$2^{[n]}\setminus\{\emptyset,[n]\}$$

can be partitioned into copies of $P$.

This conjecture concerns exact tilings after removing the bottom and top elements of the Boolean lattice, for posets that need not have unique maximal and minimal elements. The supplied text does not state that it has been resolved.