Hunter–Milojević–Sudakov–Tomon conjecture on induced Turán numbers

For positive integers $n,s$, let $K_{s,s}$ be the complete bipartite graph with $s$ vertices in each part. For a graph $H$, let $\mathrm{ex}(n,H)$ be the maximum number of edges in an $n$-vertex graph containing no copy of $H$, and 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 $H$. Hunter–Milojević–Sudakov–Tomon conjecture. For any connected bipartite graph $H$, there is a constant $C_H(s)$ depending only on $H$ and $s$ such that

$$\mathrm{ex}^*(n,H,s)\leq C_H(s)\cdot \mathrm{ex}(n,H).$$

The conjecture concerns whether forbidding a large complete bipartite subgraph and an induced connected bipartite graph changes the extremal exponent. Evidence is known for trees, cycles, the cube graph, and several classes of bipartite graphs, but the general statement remains open.