Representation of all monoids by distance-regular graphs

Determine whether distance-regular graphs can represent all finite monoids as their endomorphism monoids.

Background

After stating the open question of whether all distance-regular graphs are core-complete, the paper notes that the opposite possibility could hold: distance-regular graphs might nevertheless be sufficiently expressive to realize every finite monoid as an endomorphism monoid. The authors explicitly leave this representation question unresolved.