Conlon–Janzer–Lee conjecture for K_{2,t}-free graphs

Let $t\geq 2$ be an integer, and let $H$ be a $K_{2,t}$-free bipartite graph such that every vertex in one of the parts of $H$ has degree at most $t$. Conlon–Janzer–Lee conjecture.

$$\operatorname{ex}(n,H)=o(n^{2-1/t}).$$

This is a weaker form of the Conlon–Lee conjecture. The source presents it as an open problem at the point of formulation, although the paper's abstract states that the stronger general conjecture is proved in this work.