Erdős's conjecture on extremal odd-cycle edges

Let $G$ be an $n$-vertex graph with $e$ edges, and let a $C_{2k+1}$-edge mean an edge contained in a cycle of length $2k+1$. Assume that $e>n^2/4$, $k\geq 3$, and $n>n_k$, where $n_k$ is the threshold in the conjecture. Erdős's conjecture on extremal odd-cycle edges. If $G$ has the minimum number of $C_{2k+1}$-edges among all $n$-vertex graphs with $e$ edges, then $G$ is connected and has two blocks, one of which is a complete bipartite graph and the other of which is almost complete. The statement is presented as a corrected version of Erdős's conjecture. The surrounding discussion records asymptotic results on odd-cycle edges, but does not resolve this structural assertion.