Order isomorphisms on positive cones of non-unital $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras, not necessarily unital, and let $T:A_+\to B_+$ be a positively homogeneous order isomorphism. We first prove that $T$ is additive and extends uniquely to a bounded real-linear order isomorphism between the self-adjoint parts of $A$ and $B$. Passing to the biduals,