Existence of the asymptotic limit for universal tree graphs

Let $s(n)$ denote the minimum number of vertices in a universal graph for all trees with $n$ vertices. Asymptotic-limit conjecture. The limit

$$\lim_{n\to\infty}\frac{s(n)}{n\ln n}$$

exists. Determining this limit would answer the asymptotic version of Chung and Graham's question about $s(n)$; the stated existence remains open.