Gruslys–Leader–Tomon generalized bounded-remainder poset-packing conjecture

Let $P$ be a poset. A partition into copies of $P$ with remainder is a collection of pairwise disjoint copies of $P$ in the Boolean lattice $2^{[n]}$, together with the elements not covered by those copies.

Gruslys–Leader–Tomon conjecture. The Boolean lattice $2^{[n]}$ can be partitioned into copies of $P$ and a remainder of at most $c=c(P)$ elements.

This generalization drops the requirement that $P$ have a minimum and a maximum. The source says that proving it would resolve the earlier Gruslys–Leader–Tomon conjecture; no resolution is supplied here.