Existence of a universal vertex-condition threshold for non-rank-3 graphs

Determine whether there exists an integer t0 such that no non-rank-3 graph satisfies the t0-vertex condition.

Background

The t-vertex condition requires the numbers of subgraphs of each isomorphism type containing a specified pair of vertices to depend only on whether the pair is equal, adjacent, or non-adjacent. Rank-3 graphs satisfy this condition for every t, while the cited conjecture predicts that sufficiently high vertex conditions exclude all non-rank-3 graphs. The existence of generalized quadrangle point graphs satisfying the 7-vertex condition establishes only that any such threshold must satisfy t0 ≥ 8.