Linear polychromatic colorings of hypercube faces

A coloring of the $\ell$-dimensional faces of $Q_n$ is called $d$-polychromatic if every embedded $Q_d$ has every color on at least one face. Denote by $p^\ell(d)$ the maximum number of colors such that any $Q_n$ can be colored in this way. We provide a new lower bound on $p^\ell(d)$ for $\ell > 1$.