The rainbow triangle-packing conjecture for properly coloured complete graphs

Let $K_n$ be a complete graph properly coloured with $n$ colours. A rainbow subgraph is one whose edges have pairwise distinct colours.

Rainbow triangle-packing conjecture. The graph $K_n$ contains a rainbow subgraph that is a disjoint union of triangles covering all but at most $C$ vertices, for some absolute constant $C$.

This is the triangle-length analogue of cycle-type questions for orthomorphisms and of rainbow Hamilton-cycle conjectures. The source does not give a resolution, so the conjecture remains open.