Cambie's bounded-overlap conjecture for simultaneous edge-colourings

Let $G_1,\dots,G_k$ be graphs of maximum degree at most $\Delta$, and let $\ell$ be a positive integer such that every edge appears in at most $\ell$ of the graphs. Let $\chi'(G_1,\dots,G_k)$ be the minimum number of colours in an edge-colouring of the union whose restriction to each graph is proper.

Cambie's conjecture. Then

$$\chi'(G_1,\dots,G_k)\leq \varrho(\ell)\Delta+o(\Delta).$$

Here $\varrho(\ell)$ is the extremal asymptotic coefficient for the corresponding bounded-edge-size hypergraph colouring problem, as defined in the paper. The paper states that its results establish this conjecture asymptotically.