Almost-tiling conjecture for the Boolean lattice with copies of a poset

Let $P$ be a poset, and let $2^{[n]}$ denote the Boolean lattice of subsets of $[n]=\{1,\ldots,n\}$ ordered by inclusion. A copy of $P$ is an induced subposet isomorphic to $P$. For a set $S\subseteq 2^{[n]}$, a partition of $2^{[n]}\setminus S$ into copies of $P$ is a collection of pairwise disjoint copies whose union is that complement.

Almost-tiling conjecture. There exists a constant $c=c(P)$ such that for every positive integer $n$, there is a set $S\subseteq 2^{[n]}$ with $|S|\leq c$ such that $2^{[n]}\setminus S$ can be partitioned into copies of $P$.

This is a weaker form of Lonc's exact tiling conjecture and asserts that only a bounded number of elements need be left uncovered, independently of $n$. The supplied text does not state that this weaker conjecture has been resolved.