Füredi's conjecture for blowups of the 4-cycle

For $r\geq 3$, let

$$\mathcal{C}_4^r:=\{C_4(a,b):a+b=r,\ a,b>0\}.$$

Füredi's conjecture. The extremal number satisfies

$$\operatorname{ex}(n,\mathcal{C}_4^r)\sim \binom{n}{r-1}.$$

This conjecture concerns the extremal density of all $(a,b)$-blowups of a 4-cycle and is attributed in the paper to Füredi; no resolution is given in the supplied text.