Quadratic-threshold conjecture for near-perfect poset packings

Let $P$ be a finite poset. A $P$-packing is a family of disjoint copies of $P$. Let $c$ be an absolute constant, and consider the truncated Boolean lattice $2^{[n]}-\{\emptyset,[n]\}$.

Quadratic near-perfect packing conjecture. There exists a constant $c$ such that, if

$$n\geq c|P|^2,$$

then $2^{[n]}-\{\emptyset,[n]\}$ has a $P$-packing that covers all but at most $|P|-1$ elements.

This conjecture strengthens the paper's proved bound for the corresponding near-perfect packing result, replacing the tower-type threshold obtained in the proof by a quadratic one. It remains open.