Rainbow extension of the main bandwidth theorem

Let $k,\Delta\in\mathbb{N}$, let $\mu>0$, and let $r=1$ or $r=k$. Let $G$ satisfy the assumptions of the paper's main bandwidth theorem. An edge-colouring is locally bounded if each colour appears on at most $\nu n$ edges.

Rainbow bandwidth conjecture. There are $\nu,\beta,z>0$ and $n_0\in\mathbb{N}$ such that, whenever $G$ additionally has an edge-colouring in which every colour appears on at most $\nu n$ edges, $G$ contains rainbow copies of all graphs specified in the relevant parts of the main bandwidth theorem.

The statement is presented as a proposed generalisation of Dirac-type results to rainbow settings; the supplied text gives no resolution status.