Determine the leading coefficient for rainbow Turán numbers of long paths

Determine the correct leading coefficient of the rainbow Turán number \(\mathrm{ex}^*(n,P_\ell)\) for every path \(P_\ell\) with \(\ell\geq 6\).

Background

The paper places rainbow Turán numbers for short brooms in the broader context of rainbow Turán numbers for trees and paths. Although the ordinary Turán number of every fixed path is known asymptotically, the authors note that the rainbow analogue remains unresolved for paths of length at least six. This is an explicitly stated unresolved problem in the introductory discussion motivating the study of brooms.