On endomorphisms of alternating forms graph

A graph G is a pseudo-core if every endomorphism of G is either an automorphism or a colouring. Let Fq be the finite field with q elements and let Alt(m,q)(m≥4) be the alternating forms graph on the vector space Fqm. We prove that Alt(m,q) is a pseudo-core. Moreover, if m is odd, then Alt(m,q) is a core. If both m and q are even, then Alt(m,q) is not a core.