Robertson's Hadwiger-type conjecture for degree sequences

Let $D$ be a graphic degree sequence, and let $\mathcal{R}(D)$ be the set of its simple graph realizations. Define $\chi(D)$ as the maximum chromatic number over graphs in $\mathcal{R}(D)$, and define $H(D)$ as the maximum order of a clique minor over graphs in $\mathcal{R}(D)$. Robertson's conjecture. For every graphic degree sequence $D$,

$$\chi(D)\le H(D).$$

This is a relaxation of Hadwiger's conjecture, which asks whether every graph of chromatic number $k$ contains a $k$-clique minor. The source states that Hadwiger's conjecture remains open, while this degree-sequence relaxation is presented as a conjecture; the paper's abstract says that it is settled, but the supplied candidate does not specify the proof status of this particular claim.