Keevash–Mubayi–Sudakov–Verstraëte rainbow Turán conjecture for even cycles

Let $C_{2k}$ denote the cycle of length $2k$, and let $\mathrm{ex}^*(n,H)$ be the maximum number of edges in a properly edge-coloured $n$-vertex graph containing no rainbow copy of $H$. Keevash–Mubayi–Sudakov–Verstraëte's conjecture. For any integer $k\geq 2$,

$$\mathrm{ex}^*(n,C_{2k})=\Theta(n^{1+1/k}).$$

This conjecture asserts that the known lower bound has the correct order of magnitude for rainbow even-cycle Turán numbers; the supplied text gives no resolution beyond the lower bound.