Bukh–Conlon conjecture for powers of balanced rooted trees

Let $(T,R)$ be a balanced rooted tree, and let $T_R^\ell$ denote the graph formed from $\ell$ labeled copies of $T$ by identifying the copies on the root set $R$ and making them otherwise vertex-disjoint. For each nonempty subset $S\subseteq V(T)$, let $e_S$ be the number of edges of $T$ incident with a vertex in $S$, and define

$$\rho_T(S)=\frac{e_S}{|S|},\qquad \rho(T)=\rho_T(V(T)\setminus R).$$

The rooted tree is balanced when $\rho_T(S)\geq\rho(T)$ for every nonempty $S\subseteq V(T)\setminus R$.

Bukh–Conlon conjecture. For any balanced rooted tree $(T,R)$ and any natural number $\ell$, we have

$$\mathrm{ex}(n,T_R^\ell)=O_\ell\left(n^{2-\frac{1}{\rho(T)}}\right).$$

Bukh and Conlon proved the matching lower-bound order up to the conjectured upper bound for sufficiently large powers of any balanced rooted bipartite graph. The conjecture would therefore identify the extremal exponent for powers of balanced rooted trees and contribute to the rational exponents problem; its general status is unresolved.