Bounded-mad backbone colouring conjecture for chordal graphs

Let $q,d\in\mathbb N$. Let $G$ be a chordal graph and let $H$ be a subgraph of $G$. Write $\operatorname{Mad}(H)$ for the maximum average degree of $H$, and let $\operatorname{BBC}_q(G,H)$ denote the minimum number of colours in a colouring of $G$ such that endpoints of every edge of $H$ receive colours differing by at least $q$. Then there exists a function

$$f:\mathbb N^2\to\mathbb N$$

such that the following holds for every such $G$ and $H$ with $\operatorname{Mad}(H)\leq d$.

Bounded-mad backbone colouring conjecture.

$$\operatorname{BBC}_q(G,H)\leq \omega(G)+f(q,d).$$

This is stated as a stronger conjecture than the spanning-forest case, since forests have bounded maximum average degree. The source says that the conjecture is believed to hold; no resolution is given.