Largest t-intersecting families for t up to n/2

Prove that for every positive integer t with t ≤ n/2, the largest t-intersecting family of spanning trees of K_n consists of all spanning trees containing a fixed set of t pairwise disjoint edges.

Background

The main theorem establishes this characterization only when n is sufficiently large relative to t, specifically for t bounded by a constant multiple of n/log n. The authors conjecture that the same extremal structure remains valid throughout the full range t ≤ n/2, where a forest of t disjoint edges can exist.