Asymptotic behavior for joint Turán number for consecutive cycles

For a finite family \( {\mathcal F}\) of graphs, let \( t(n, {\mathcal{F}})\) denote the smallest integer \( m\) that every graph on \( n\) vertices and \( m\) edges must contain a member of \( \mathcal{F}\) as a subgraph. Erdös and Simonovits [1] raised the following question:

Erdös and Simonovits [1] proved that \( t(n,\{C_4,C_5\}) = \frac{1}{2\sqrt 2} n^{3/2}+O(n)\).

A problem on the Turán number for \( C_{2k-1} \) and \(C_{2k}\) (proposed by Erdös and Simonovits [1])

Is it true that \[ t(n,\{C_{2k-1},C_{2k}\}) = (1+o(1)) (\frac n 2)^{1+1/k}? \]

Bibliography
1 P. Erdös and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275-288.