Existence of Bukh–Conlon densities in every fractional class

For a rooted graph $F$, let $R(F)$ be its root set and define its density by

$$\rho_F=\frac{e(F)}{v(F)-|R(F)|}.$$

Call $F$ balanced if $\rho_F>1$ and every subset $S\subseteq V(F)\setminus R(F)$ is incident with at least $\rho_F|S|$ edges. A fraction $b/a$ is a Bukh–Conlon density if there is a balanced rooted tree $F$ with $\rho_F=b/a$ such that, for every $p\in\mathbb{N}^+$,

$$\operatorname{ex}(n,F^p)=O\left(n^{2-1/\rho_F}\right).$$

Existence of Bukh–Conlon densities in every fractional class. For every $s,a\in\mathbb{N}$ with $s<a$, there exists $m\in\mathbb{N}^+$ such that $m+s/a$ is a Bukh–Conlon density.

This is a weaker conjecture proposed after describing known constructions of Bukh–Conlon densities and remaining examples such as $m+4/7$. The source gives no resolution status.