Multiple log-concavity violations in a tree

Construct or identify trees whose independent set sequences have log-concavity broken at more than one index.

Background

For the family of trees $T_{m,t}$ constructed in the paper, the authors prove that log-concavity is broken at $mt+2$ and explain that it cannot be broken at any later index. They further report that all their computations show either log-concavity throughout or a single violation at $mt+2$. This motivates the unresolved question of whether some trees can exhibit log-concavity failures at multiple positions in their independent set sequences.