Conlon's conjecture on the extremal number of fixed-length tight cycles

Let $r\geq 2$ and let $C^{(r)}_{\ell}$ denote the $r$-uniform tight cycle of length $\ell$. Conlon's conjecture. There exists $c=c(r)>0$ such that, for every $\ell\geq r+1$ divisible by $r$,

$$\operatorname{ex}(n,C^{(r)}_{\ell})=O\left(n^{r-1+\frac{c}{\ell}}\right).$$

This is a conjecture about the extremal number of tight cycles of each fixed admissible length. The supplied text gives no resolution status.