Extent of early log-concavity failure in trees

Determine whether there exists a constant c>0 such that trees with arbitrarily large independence number have independent set sequences whose log-concavity is broken at an index no later than approximately (1-c) times their independence number.

Background

The paper constructs trees whose independent set sequences violate log-concavity at a position approximately α(T)/(16log⁡α(T))\alpha(T)/(16\log \alpha(T)) before the final index, thereby confirming a conjecture of Kadrawi and Levit that violations can occur arbitrarily far from the end of the sequence. The authors then ask how much farther toward the beginning such a violation can be pushed. Specifically, the unresolved issue is whether violations can occur at a linear distance from the maximum independent-set size, rather than only at the logarithmically smaller scale established by the construction.

References

We do not know how far further beyond \Theta\left(\alpha(T)/\log \alpha(T)\right) we can push the point where log-concavity can be broken.

Is there a c>0 such that there are trees T with arbitrarily large independence number \alpha(T), for which the log-concavity of the independent set sequence of T is broken at no later than around (1-c)\alpha(T)?

— Trees with non log-concave independent set sequences  (2502.10654 - Galvin, 15 Feb 2025) in Section 3, Discussion