An infinite family of intransitive directed strongly regular graphs with rank 6 Weisfeiler--Leman closure

We construct an infinite family of directed strongly regular graphs (DSRGs) with intransitive full automorphism groups whose Weisfeiler–Leman closure are association schemes of rank 6. This is the smallest possible rank for an association scheme admitting a proper DSRG merging. We also exhibit rigid sporadic DSRGs with the same closure property, showing that combinatorial regularity, group-theoretic symmetry, and Weisfeiler–Leman regularity capture fundamentally different aspects of graph structure.