The many-colour edge-density conjecture for non-monochromatic triangles

Let $G=(V,E)$ be a $k$-edge-coloured graph with colouring

$$E=\bigsqcup_{i=1}^k E_i.$$

Many-colour edge-density conjecture. If

$$2|E|-\max\{|E_i|\,:\,1\leq i\leq k\}>\frac{1}{2}|V|^2,$$

then $G$ contains a non-monochromatic triangle.

This conjecture proposes a many-colour extension of the paper's theorem and corollary on non-monochromatic triangles. Its resolution is not given in the supplied text.