The refined bounded-degree conjecture for full rainbow sets

Let $f_{\mathcal D(k)}(n)$ denote the full-rainbow parameter for the class $\mathcal D(k)$ of graphs with maximum degree at most $k$, and let $q$ be the quantity used in the source's preceding discussion. Refined bounded-degree conjecture. For $k>2$ and $n>3$,

$$f_{\mathcal D(k)}(n)=q(n-1)+1.$$

The statement appears in the paper's summary of open problems, but the supplied context does not define $q$ or explicitly identify $f_{\mathcal D(k)}(n)$ with the two-parameter notation.