Grzesik–Janzer–Nagy conjecture for blow-ups of even cycles

For a graph $H$, let $\mathrm{ex}(n,H)$ denote its Turán number, and let $C_{2k}[r]$ be the $r$-blow-up of the cycle $C_{2k}$.

Grzesik–Janzer–Nagy conjecture for even cycles. For every pair of integers $r,k\geq2$,

$$\mathrm{ex}(n,C_{2k}[r])=O\left(n^{2-\frac{1}{r}+\frac{1}{rk}}\right).$$

This is the specialization of the general blow-up conjecture to even cycles and is open for $k\geq3$; the case $k=2$, corresponding to $C_4$, is covered by the known complete-bipartite-graph case.