Rainbow Turán bound for cycles

Prove that every n-vertex properly edge-coloured graph with no rainbow cycles has at most O(n log n) edges.

Background

The hypercube Q_d shows the lower bound Ω(n log n). Janzer, Tomon, and later Janzer–Sudakov and Kim–Lee–Liu–Tran proved upper bounds O(n (log n)^4), O(n (log n)^{2+o(1)}), and O(n (log n)^2). The conjectured O(n log n) bound remains open and would be tight up to constants.