Rational exponent conjecture for degenerate hypergraph Turán numbers

Let $r\ge 3$, and let $\mathcal{F}$ be a degenerate finite family of $r$-graphs satisfying

$$ex{ex}(n,\mathcal{F})=\Omega(n^{1+\alpha})$$

for some constant $\alpha>r-2$. Then there exist constants $\beta>0$, $c>0$, and $C>0$ such that, for all sufficiently large $n$,

$$c\le \frac{ex{ex}(n,\mathcal{F})}{n^{1+\beta}}\le C.$$

This proposed hypergraph analogue of the Erdős–Simonovits rational exponent conjecture would imply the claimed exponent tightness for the subcritical range $p<1/(r-1-\alpha)$. Its status is open; the paper presents it as a bold conjecture rather than proving it.