The rainbow Turán path conjecture

Let $P_{\ell}$ be the path with $\ell$ edges, and let $\operatorname{ex}^*(n,P_{\ell})$ denote the maximum number of edges in a properly edge-colored $n$-vertex graph containing no rainbow copy of $P_{\ell}$. Here $O(1)$ denotes a quantity bounded independently of $n$.

Rainbow Turán path conjecture. For all $\ell \geq 3$,

$$\operatorname{ex}^*(n,P_{\ell}) = \frac{\ell}{2}n + O(1).$$

The lower bound is supplied by Johnston and Rombach's construction, while the cases $\ell=3,4$ were known and the paper proves the matching asymptotic upper bound for $P_5$. The assertion remains open in general for longer paths.