Problem #809: Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$...

Let $k\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\lfloor n^2/4\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges.

Is it true that\[F_k(n)\sim n^2/8?\]