Kadrawi–Levit conjecture on the locations of log-concavity breakdowns in tree independence sequences

Let $T$ be a tree, let $\alpha(T)$ denote its independence number, and let $(a_i)_{i=0}^{\alpha(T)}$ be its independent set sequence. Log-concavity is broken at $k$ when $a_k^2<a_{k-1}a_{k+1}$. Kadrawi–Levit's conjecture. For every $\ell\geq 1$, there is a tree $T=T(\ell)$ for which log-concavity of the independent set sequence is broken at $\alpha(T)-\ell$. This conjecture asks how far from the end of a tree's independent set sequence log-concavity can fail; earlier examples broke log-concavity only at the last possible place, while an ad hoc example broke it at $\alpha(T)-2$.