Erdős's balanced blow-up conjecture for 5-cycles

For $n\geq5$, consider the balanced blow-up of the $5$-cycle $C_5$, obtained by replacing each vertex of $C_5$ by an independent set of size $\lfloor n/5\rfloor$ or $\lceil n/5\rceil$, and each edge by a complete bipartite graph between the corresponding sets. Erdős's conjecture. This balanced blow-up maximizes the number of $5$-cycles among all triangle-free graphs of order $n$. The claim is an extremal refinement of the preceding bound; the source gives no resolution.