The maximum product of the Sombor and modified Sombor indices of a tree

The Sombor index and its modified counterpart weight each edge of a graph by the Euclideannorm of its endpoint degrees and the reciprocal of that norm. Their product measures thevariation of those weights across the graph, and its maximum over trees is not determined bymaximizing either factor separately. Gutman, Redžepović, and Furtula conjectured in 2023that an 𝑛-vertex tree maximizing this product is a specific eclipsed sun. We prove their sharpinequality and the uniqueness of the extremizer for every 𝑛 ≥ 5. The proof first excludes treeswithout a dominant-degree vertex, then compares the remaining trees directly with eclipsedsuns by a nonnegative edge defect, and finally obtains the optimal arm count by signed finitedifferences. The infinite families are treated analytically; finite exceptional ranges are resolvedby exact rational intervals and enumeration of unlabeled trees. The result settles the statedextremal-tree conjectures, not the unrelated questions concerning chemical correlations of theindex.