Consider the action $L=\chi\square^3\chi$
On one hand, Ostrogradski's theorem seems to indicate that it is not unitary.
On the other hand, it can be reached by replacing $\phi$ with $\square \chi$ in KleinGordon action.
Namely, KleinGordon Action is well known to be healthy. Now if we replace in it with $\phi=\square\chi$, we get the action $L=\chi\square^3\chi$.
Does this make the Klein Gordon Action unhealthy?