Linear lower bound for additive path-length sums

Prove that there exists a constant \(\alpha>0\) such that, for every integer \(n\geq 1\) and every sequence of non-negative integers \((a_i)_{i=1}^n\) satisfying \(a_i\leq \alpha n\), the set \(\{a_i+a_j+(j-i):1\leq i<j\leq n\}\) has cardinality at least \(\alpha n\).

Background

This conjecture arises from the authors’ analysis of short leaf-to-leaf path lengths. In the proof of their lower bound, path lengths are represented through expressions involving two auxiliary depths $a_i,a_j$ and the separation $j-i$ along a long path.

The conjecture asserts a linear number of distinct values for these additive expressions whenever all sequence entries are at most a linear fraction of $n$. The paper proves only a weaker $n^{2/3}$-scale statement and notes that the conjecture would imply the short-length conjecture for 1–3 trees.