Jovović–Koledin–Stanić spectral-gap conjecture for trees

Let $\mathcal{T}(n)$ be the family of trees on $n$ vertices. For a tree, write $\lambda_1$ and $\lambda_2$ for its two largest adjacency eigenvalues, and let $DC(k,k,\ell)$ denote a double comet with two equal pendant-path lengths $k$ and central path length $\ell$, where $2k+\ell=n$. Jovović–Koledin–Stanić conjecture. In $\mathcal{T}(n)$, the spectral gap $\lambda_1-\lambda_2$ is minimized by a double comet $DC(k,k,\ell)$ such that $2k+\ell=n$. The conjecture concerns the extremal structure of the spectral gap among trees and is part of the broader study of combinations of the first two eigenvalues. The supplied context gives no resolution, so its status remains open.