The extremal number conjecture for powers of balanced rooted trees

Let $(T,R)$ be a balanced rooted tree, let $T^p$ be the graph formed from the union of $p$ distinct labelled copies of $T$ that agree on the roots $R$ and are otherwise disjoint, and let $\rho_T$ denote the rooted density of $T$. Extremal number conjecture. For any balanced rooted tree $(T,R)$,

$$\operatorname{ex}(n,T^p)=O_p\left(n^{2-1/\rho_T}\right).$$

The preceding lower-bound result gives $\operatorname{ex}(n,T^p)=\Omega(n^{2-1/\rho_T})$ for balanced rooted trees, so this conjecture would determine the order of magnitude of the extremal number. It is posed as the corresponding upper bound and remains open in the source.