Conjecture on cycles in odd-cycle-free graphs

Let $k>1$, and let $C_{2k+1}$ denote the cycle of length $2k+1$. A graph is $C_{2k+1}$-free if it contains no subgraph isomorphic to $C_{2k+1}$; $K_{a,b}$ denotes the complete bipartite graph with parts of sizes $a$ and $b$. Odd-cycle-free cycle conjecture. For any $k>1$, if an $n$-vertex $C_{2k+1}$-free graph $G$ has the maximum number of cycles, then

$$G=K_{\left \lceil \frac{n}{2}\right \rceil,\left \lfloor \frac{n}{2}\right \rfloor}.$$

The conjecture extends the extremal cycle question from triangle-free graphs to longer odd-cycle exclusions. The surrounding discussion notes related extremal edge and maximal-graph results, but the conjecture's resolution is not supplied.