The association-scheme conjecture for the meet-table graphs on 12 points

Let $\mathcal{U}^{12}_3$ be the set of uniform $3$-partitions of a $12$-set. For each relevant meet-table isomorphism type, let $G_i$ be the graph on $\mathcal{U}^{12}_3$ defined in the paper by adjacency of partitions with that meet-table type.

Association-scheme conjecture for 12 points. The graphs

$$\{G_i:i=2,\dots,9\}$$

form an association scheme on $\mathcal{U}^{12}_3$.

If true, the scheme would justify the ratio bound used in the paper and imply the stated upper bound on the clique number of $UQI(12,3)$.