On the restricted homomorphism problem

The restricted homomorphism problem  RHP(H,Y) asks: given an input digraph GG and a homomorphism g:G→Yg:G→Y, does there exist a homomorphism f:G→Hf:G→H? We prove that if HH is hereditarily hard   and Y↛HY↛H, then RHP(H,Y) is NP-complete. Since non-bipartite graphs are hereditarily hard, this settles in the affirmative a conjecture of Hell and Nešetřil.