Bipartition-respecting packing coloring conjecture for hypercubes

Let $Q_n$ be the $n$-dimensional hypercube. A packing coloring respects bipartition if all vertices assigned color $1$ lie in one part of the bipartition. Let $\chi_\rho(Q_n)$ denote the packing chromatic number of $Q_n$, and let $\chi_\rho^B(Q_n)$ denote the minimum number of colors in a packing coloring of $Q_n$ that respects its bipartition.

Bipartition-respecting packing coloring conjecture. For every $n\geq 0$,

$$\chi_\rho(Q_n)=\chi_\rho^B(Q_n).$$

The invariant $\chi_\rho^B(Q_n)$ is an upper bound for the packing chromatic number, and the conjecture asserts that requiring color $1$ to be contained in one bipartition class does not increase the minimum number of colors. The exact value of $\chi_\rho(Q_9)$ is still undetermined, so the conjecture concerns the relationship between these two parameters rather than a currently known exact value.