Dehmer–Kraus's two-tailed-comet conjecture for tree entropies

Let $T_{n,d}$ be the family of trees of order $n$ with fixed diameter $d$, and assume $d\ll n$. Let $C_{n,\lfloor n/2\rfloor,\lceil n/2\rceil}$ denote the two-tailed comet appearing in the claim. The entropies under consideration are $I_{f^2}(G)$ and $I_{f^3}(G)$.

Dehmer–Kraus's conjecture. Among all trees $T_{n,d}$ with $d\ll n$, the two-tailed comet

$$C_{n,\lfloor n/2\rfloor,\lceil n/2\rceil}$$

achieves the maximal values of the entropies $I_{f^2}(G)$ and $I_{f^3}(G)$.

The preceding theorem proves a comparison between two-tailed comets and ordinary comets for specified decreasing weight sequences, whereas this conjecture asserts a broader extremal property among all trees in the family. The source gives no resolution.