Convexity of coherent information - erroneous argument!

Consider a state $rho_{AB}$. Let it have purification $psi_{A’AB}$. I am interested in the coherent information of this state which is given by

$$I(Arangle B)_rho = S(B)_rho – S(AB)_rho$$

I can rewrite this as follows using the fact that $psi_{A’AB}$ is pure.
begin{align}
I(Arangle B) &= S(B)_rho – S(AB)_rho
&=S(A’A)_psi – S(A’)_psi
&= S(A|A’)_psi
end{align}

Now let us consider $J(Arangle B) = -I(Arangle B)$. We have
begin{align}
J(Arangle B) &= S(AB)_rho – S(B)_rho
&= S(A|B)_psi
end{align}

But the quantum conditional entropy $S(X|Y)$ is a concave function of the input state (see Corollary 11.13 in Nielsen and Chuang). So both $J$ and $I$ are concave. This would imply that the coherent information is a linear function of $psi_{A’AB}$ which I believe is not true. Where did I make a mistake in the argument?