Rao–Rao $k$-factor conjecture for graphic degree sequences

Let $\pi_1$ be a graphic degree sequence, and let $\mathbf{k}$ denote the degree sequence in which every term is $k$. Two graphic degree sequences pack when they have realizations on the same vertex set with edge-disjoint edge sets. Rao–Rao $k$-factor conjecture. Graphic degree sequences $\pi_1$ and $\mathbf{k}$ pack if and only if $\pi_1+\mathbf{k}$ is graphic. This generalizes Grünbaum's conjecture for the all-ones sequence and asks when the necessary condition that the componentwise sum be graphic is also sufficient; the source provides no resolution status.