Burr–Erdős–Graham–Sós maximal anti-Ramsey conjecture for odd cycles

Let $f(n,e,H)$ be the minimum number of colors in an edge-coloring of an $n$-vertex graph with at least $e$ edges such that every copy of $H$ is rainbow. Let $C_{2k+1}$ denote the cycle of length $2k+1$.

Burr–Erdős–Graham–Sós conjecture. For every integer $k\ge 3$,

$$f\left(n,\left\lfloor \frac{n^2}{4} \right\rfloor+1,C_{2k+1}\right)=\frac{n^2}{8}+o(n^2).$$

The conjecture asks for the precise quadratic asymptotics of the maximal anti-Ramsey function just above the Turán number of an odd cycle. The corresponding function is constant for triangles, linear for $5$-cycles, and quadratic for all longer odd cycles; the conjectured leading constant remains open according to the supplied status evidence.