Rainbow complete-graph subdivision conjecture

A proper edge-colouring is one in which incident edges receive distinct colours, and a rainbow $K_k$-subdivision is a subdivision of $K_k$ whose edges have pairwise distinct colours. Rainbow-subdivision conjecture. For each $k\geq 3$, every properly-coloured $n$-vertex graph with no rainbow $K_k$-subdivision has $O(n\log n)$ edges. The source presents this as a natural generalisation of the rainbow-cycle conjecture and gives no resolution.