DeBiasio–Kamel–McCourt–Sheats bounded-diameter extension of Ryser's conjecture

Let $\alpha\geq 1$, let $r\geq 2$, and let $G$ be a graph with independence number $\alpha(G)=\alpha$. An $r$-colouring of $G$ assigns one of $r$ colours to each edge, and a monochromatic component is a connected component in one colour class.

DeBiasio–Kamel–McCourt–Sheats conjecture. There exists a constant $d=d(\alpha)$, depending only on $\alpha$, such that every $r$-colouring of $G$ has $(r-1)\alpha$ monochromatic components of diameter at most $d$ whose vertex sets cover $G$.

This is a bounded-diameter extension of Ryser's conjecture from complete graphs to graphs with prescribed independence number. The source presents it as open; it would imply the corresponding bounded-diameter cover for every fixed independence number, uniformly in $r$ and in the graph.