Dong–Gao–Li–Liu induced rational exponents conjecture

For positive integers $n,s$, let $K_{s,s}$) be the complete bipartite graph with $s$ vertices in each part. For a bipartite graph $H$, let $\mathrm{ex}^*(n,H,s)$ be the maximum number of edges in an $n$-vertex graph containing neither a copy of $K_{s,s}$ nor an induced copy of $H$. Dong–Gao–Li–Liu induced rational exponents conjecture. For every rational number $r\in(1,2)$, there exist a bipartite graph $H$ and a constant $s_0$ such that

$$\mathrm{ex}^*(n,H,s)=\Theta_s(n^r)$$

for any $s\geq s_0$. This is the induced analogue of the rational exponents conjecture for ordinary Turán numbers. The source reports a proof for all $r=2-a/b$ with $a,b\in\mathbb{N}$ and $b\geq\max\{a,(a-1)^2\}$, while the full range of rational exponents in $(1,2)$ remains open.