# Schwarz inequality for unital positive maps on C*-algebras

I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that

Let $$\phi : \mathcal{A} \to \mathcal{B}$$ be a unital positive linear map between the two C*-algebras $$\mathcal{A}, \mathcal{B}$$. Then

$$\phi(A^* A) \ge \phi(A)^* \, \phi(A)$$ and

$$\phi(A^* A) \ge \phi(A) \, \phi(A)^*$$

for every subnormal $$A \in \mathcal{A}$$.

He then conjectures that the same result might apply also for hyponormal opertators, i.e. operators such that $$A^* A \ge A A^*$$ (this conjecture is equivalent to the Woronowicz's conjecture).

I was wondering whether this conjecture has been better investigated or not. Also, is there a characterization of hyponormal operators?