Bradač–Janzer–Sudakov–Tomon conjecture for Cartesian products of trees

For graphs $F$ and $G$, their Cartesian product $F\Box G$ has vertex set $V(F)\times V(G)$, with $(u,v)$ adjacent to $(u',v')$ if and only if either $u=u'$ and $vv'\in E(G)$, or $v=v'$ and $uu'\in E(F)$. Let $T$ and $S$ be trees, each with at least one edge. Bradač–Janzer–Sudakov–Tomon conjecture. There exist positive real numbers $c$ and $C$ such that

$$cn^{3/2}\leq \mathrm{ex}(n,T\Box S)\leq Cn^{3/2}.$$

This extends the proved order of magnitude for the Cartesian product of a nontrivial tree and a nontrivial path. The conjecture remains open in general.