Induced rational exponents near two

Given a bipartite graph $H$ and a natural number $s$, let ${\mathrm{e}\mathrm{x}}^{\ast }(n,H,s)$ denote the maximum number of edges in an $n$-vertex graph that contains neither $K_{s,s}$ nor an induced copy of $H$. Hunter, Milojević, Sudakov, and Tomon conjectured that ${\mathrm{e}\mathrm{x}}^{\ast }(n,H,s)=O_{H,s}(\mathrm{e}\mathrm{x}(n,H))$ whenever $H$ is connected. Motivated by this conjecture and the rational exponents conjecture, Dong, Gao, Li, and Liu conjectured that for every rational $r\in (1,2)$ there is a bipartite graph $H$ and an $s_0$ such that ${\mathrm{e}\mathrm{x}}^{\ast }(n,H,s)=\mathrm{\Theta }(n^r)$ for all $s\ge s_0$. We prove that the latter conjecture holds for all rationals $r=2-a\mathrm{/}b$, where $a,b\in \mathbb{N}$ satisfy $b\ge \max \{a,(a-1{)}^2\}$. Our result extends a well-known result of Conlon and Janzer to the induced setting and adds more evidence to support the former conjecture.