A logarithmic upper bound for the moment-to-Perron-value ratio of rooted trees

Let $T$ be a rooted tree of order $n$, and let $\rho(T)$ denote its Perron value and $\mu(T)$ its moment. Logarithmic bound conjecture. There exists $\alpha_0\in\mathbb{R}$ such that

$$\mu(T)\leq \alpha_0\ln(n)\rho(T)$$

for every rooted tree $T$ of order $n$. The preceding constructions show that the ratio $\mu(T)/\rho(T)$ is unbounded, while suggesting that its growth is at most logarithmic in the order of the tree.