Purely combinatorial proof of the four-vertex matching-index bound

Construct a short purely combinatorial proof that every graph on four vertices has matching index at most 3.

Background

The Krenn–Gu conjecture can be reduced, in part, to structural statements about smaller graphs. Prior work used a computer-assisted result of Mantey establishing that every graph on four vertices has matching index at most 3.

Although the four-vertex bound is known computationally, the paper explicitly identifies the absence of a short purely combinatorial proof as unresolved. This problem concerns the proof method rather than the truth of the bound itself.