Xu–Zhang–Jing–Ge conjecture on colour-isomorphic even cycles

For an integer $r\geq 2$ and a graph $H$, let $f_r(n,H)$ be the smallest number $C$ such that some proper edge-colouring of $K_n$ with $C$ colours contains no $r$ vertex-disjoint colour-isomorphic copies of $H$. Xu–Zhang–Jing–Ge's conjecture. For any $k\geq 3$,

$$f_2(n,C_{2k})=\Omega(n^{2-\frac{2}{k}}).$$

This refines an open problem of Conlon and Tyomkyn about lower bounds approaching $n^2$ for colour-isomorphic even cycles. The supplied text states that the conjecture is proved later in the paper in a more general form.