Concatenation of two strong hashes may have striking weakness
If we use $H_1(X) = H_0(X) \oplus firstnbits(X)$, this would seem to be trivial.

EDIT: As Cédric Van Rompay pointed out, this is only a counterexample if $H_1$ winds up being preimage-resistant. This may not be a necessary consequence of $H_0$ being preimage-resistant, but I really only need one case where it is.