The association-scheme conjecture for the meet-table graphs on 15 points

Let $\mathcal{U}^{15}_3$ be the set of uniform $3$-partitions of a $15$-set. For each relevant meet-table isomorphism type, let $G_i$ be the graph on $\mathcal{U}^{15}_3$ defined in the paper by adjacency of partitions with that type.

Association-scheme conjecture for 15 points. The graphs $G_i$, for $i=2,\dots,13$, form an association scheme on $\mathcal{U}^{15}_3$.

In this case the qualitative independence relation is represented by three graphs in the proposed scheme rather than by a single graph, motivating this conjecture.