Core-completeness of distance-regular graphs

Determine whether every distance-regular graph is core-complete, meaning that every endomorphism of the graph is an automorphism or has a complete graph as its image.

Background

The paper observes that strongly regular graphs are core-complete, so their endomorphisms are too restricted for strongly regular graphs to represent all finite monoids. It then asks whether this property extends from strongly regular graphs to the broader class of distance-regular graphs. The question concerns the structural classification of endomorphisms of distance-regular graphs.