Rainbow-cycle Turán conjecture

A proper edge-colouring is one in which incident edges receive distinct colours, and a rainbow cycle has pairwise distinct edge colours. Rainbow-cycle conjecture. Every properly-coloured $n$-vertex graph with no rainbow cycle has $O(n\log n)$ edges. The best bound described in the source is $O(n\log n\log\log n)$, and the source notes doubts about the conjecture's truth.