Improving the lower bound to n^{0.9}

Prove or disprove that the largest common subtree size X_n of two independent uniform attachment trees satisfies X_n\geq n^{0.9} with high probability.

Background

The paper establishes X_n\geq n{0.83} with high probability by applying local subtree-switching optimizations to the Ulam–Harris common subtree. Simulations suggest an exponent near 0.9 for a root-to-root matching, but no corresponding rigorous lower bound for the unrestricted largest common subtree is proved.

The authors explicitly pose n{0.9} as a possible next benchmark and note that it is conditional on the statement being true.

References

For instance, can one prove that, say, $X_{n} \geq n{0.9}$ with high probability (if this is true)?

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “Asymptotics of X_n”

It seems plausible that further local optimizations (beyond those in Section~\ref{sec:beyondUH}) could improve upon Theorem~\ref{thm:main} to a certain extent, but it is unclear if these methods could be pushed as far as to give a lower bound of, say, $n{0.9}$.

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “Asymptotics of X_n”