Exponent improvement conjecture for even prisms

For each integer $k\geq2$, consider the Cartesian product $C_{2k}\square P_2$ of the cycle $C_{2k}$ and the path $P_2$. The notation $\operatorname{ex}(n,H)$ denotes the maximum number of edges in an $n$-vertex graph containing no copy of $H$. Even-prism exponent improvement conjecture. For every $k\geq2$, there exists $\varepsilon=\varepsilon(k)>0$ such that

$$\operatorname{ex}(n,C_{2k}\square P_2)=O\left(n^{\frac53-\varepsilon}\right).$$

This would improve the currently stated general upper bound $\operatorname{ex}(n,C_{2k}\square P_2)=O(n^{5/3})$; the paper presents the improvement as an open conjecture.