Optimal union vertex-distinguishing edge colorings of forests

Let $F$ be a forest on $n$ vertices, and let $\chi'_{\cup}(F)$ denote its union vertex-distinguishing edge chromatic number, where edge colors are sets and the union of the colors on the edges incident with a vertex distinguishes every pair of vertices. Forest optimal-coloring conjecture.

$$\chi'_{\cup}(F)=\left\lceil \log_2\left(n +1\right) \right\rceil.$$

The paper has established that this value is one of two consecutive possibilities for general graphs and notes that optimal coloring is open even for trees; the conjecture asserts the lower value for all forests.