Goubko's BFS-tree conjecture for terminal distance spectral radius

Let $\mathbf{d}$ be a tree degree sequence, and let $BFS(\mathbf{d})$ denote its breadth-first-search (greedy) tree. For a graph $G$, let $RD(G)$ be its terminal distance matrix, and let the terminal distance spectral radius be the spectral radius of $RD(G)$. The BFS-tree conjecture for terminal distance spectral radius. The tree $BFS(\mathbf{d})$ has the minimum terminal distance spectral radius among all trees with degree sequence $\mathbf{d}$. The paper establishes lower and upper bounds whose relative gap is numerically at most $3\%$, but the exact minimum statement remains open.