The fixed-diameter extremal tree conjecture for distance entropies

Let $T$ be a tree of order $n$ and diameter $d\ll n$. Let $I_{f_s}(T)$ and $I_{f_e}(T)$ be the distance-based entropies associated with the functionals $f_s$ and $f_e$. The fixed-diameter extremal tree conjecture. The maximum values of $I_{f_s}$ and $I_{f_e}$ are attained by the tree obtained by identifying the central vertex of the star of order $n-d$ with a central vertex of the path of length $d$. The paper explains that this conjecture is false for general $I_{f_e}$, since the coefficients can be chosen so that an arbitrary preferred tree is extremal; it is true for $I_{ecc}$ in the relevant specialization.