Conjecture on endomorphisms of antipodal non-bipartite distance-regular graphs

Let $X$ be an antipodal, non-bipartite distance-regular graph of diameter $3$. An endomorphism of $X$ is a graph homomorphism from $X$ to itself, and a subgraph has diameter $2$ when its greatest distance between vertices is $2$. Endomorphism conjecture. The graph $X$ has no endomorphism to a subgraph with diameter $2$. The conjecture is motivated by computations for feasible intersection arrays of primitive distance-regular graphs of diameter $3$, which found no such examples among graphs of degree at most $50$; whether the assertion holds in general remains open.