The bounded-degree rainbow independent-set conjecture

Let $\mathcal D(k)$ be the class of graphs whose vertex degrees are at most $k$. Bounded-degree conjecture.

$$f_{\mathcal D(k)}(n,n)=\left\lceil\frac{k+1}{2}\right\rceil(n-1)+1.$$

The source gives a general upper bound from graph colouring and says that this proposed value is not known to be sharp in general.