How fast can we find and disconnect roots in a forest?
The problem has name "fringe marked ancestor problem" and indeed has $O(\log \log n)$ worst-case solution for both operations [1], thus overcoming the lower bound for generic version of the problem. Their solution is based on Euler tour of the tree with union-split-find structure (and fast LCA for trees with unbounded degree).

The same paper states that it is an open problem whether this bound is tight.

[1] D. Breslauer and G. F. Italiano. Near real-time suffix tree construction via the fringe marked ancestor problem. Selected papers from the 18th International Symposium on String Processing and Information Retrieval (SPIRE 2011) https://www.sciencedirect.com/science/article/pii/S1570866712001062#br0230