Grzesik–Janzer–Nagy conjecture on Turán numbers of graph blow-ups

Let $F$ be a graph, and let $F[r]$ denote its $r$-blow-up, obtained by replacing each vertex of $F$ with an independent set of size $r$ and each edge with a copy of $K_{r,r}$. Let $\mathrm{ex}(n,F)$ be the maximum number of edges in an $F$-free graph on $n$ vertices.

Grzesik–Janzer–Nagy conjecture. If

$$\mathrm{ex}(n,F)=O(n^{2-\alpha})$$

for some constant $0\leq\alpha\leq1$, then for every positive integer $r$,

$$\mathrm{ex}(n,F[r])=O\left(n^{2-\frac{\alpha}{r}}\right).$$

The conjecture is known when $F$ is a tree and when $F=K_{s,t}$ with $\alpha=1/s$, including the case $C_4=K_{2,2}$, but it remains open already for even cycles $F=C_{2k}$ with $k\geq3$.