are locally unique pure quantum states also ground states of some local hamiltonian?

Let $H=\sum_i H_i$ be some k-local hamiltonian with a unique ground state $|\psi>$. Then it is easily shown that $|\psi>$ is k-locally distinguishable from any other state $|\psi'>$.

Is the converse also true?

In other words assume $|\psi>$ is a pure state such that for any other pure state $|\psi'>$ there exists a subset of qubits $K$ s.t $|K| \leq k$ for a fixed $k>1$ and

$tr_{[n]\backslash K}(|\psi>) \neq tr_{[n]\backslash K}(|\psi'>)$

is it true that there exists some hamiltonian $H=\sum_i H_i$ where $H_i$ acts on at most k qubits s.t $|\psi>$ is the only ground state of $H$?