Erdős–Faudree–Rousseau conjecture on edges occurring in odd cycles

Let $G$ be an $n$-vertex graph with

|E(G)|=\left\left\oldlfloor\frac{n^2}{4}\right\right\oldrfloor+1.

Let $C_{2k+1}$ denote the cycle of length $2k+1$, and say that an edge occurs in $C_{2k+1}$ if it belongs to a copy of that cycle. Fix an integer $k\ge 2$. Erdős–Faudree–Rousseau conjecture. Every such graph contains at least

$$\frac{2}{9}n^2-O(n)$$

edges that occur in $C_{2k+1}$. This conjecture asserts that the construction consisting of a complete graph and a complete balanced bipartite graph as two $2$-connected blocks overlapping in one vertex is extremal up to the linear error term. The case $k=2$ was identified as an open problem in the cited work; no resolution is supplied here.