Consider the Hamiltonian
$$H=p^2+ix^3+ix.$$
This paper by Carl M bender claims this is a $PT$ symmetric Hamiltonian. In this he describes $PT$ symmetry as
parity $P$, whose effect is to make spatial reflections, $$p\to-p\text{ and }x\to -x$$ time reversal $T$, which replaces $$p\to -p,\ x\to-x\text{, and }i\to-i.$$
If I carry out these operations, the Hamiltonian $H$ is only invariant under time reversal $T$ and not invariant under $P$ (spatial reflections).
Is there any other way to check if a Hamiltonian is $PT$ symmetric? Please illustrate with the above example.