if you don't tag me I don't get a notification

@user1936752 anyway, I see that this is a problem of terminology. You are referring to a product state obtained from the maximally mixed state on the individual systems, that is, up to normalisation, $I\otimes I$. Then yes, this is a product state gives no correlations, and indeed you get $J=0$. It also has no bearing with the question, and nowhere in the answer I mentioned this state

nor does it give the same value of the correlation as the maximally entangled state, so I don't understand what is your objection

@gIs

I have no objection to your answer and I even upvoted it. I initially thought J(\rho) was zero for maximally mixed state and then corrected myself in my comment because my very first comment was wrong. Also, I suspect there is one thing that you might be misunderstanding which is that the identity matrix in $H_A\otimes H_B$ is always the product of individual identities. $diag(1,1,..)$ on a joint space is always $diag(1,1,..)\otimes diag(1,1...)$

@gIS

Meant to say that in my first comment I wrongly wrote J(rho) = 1 for max mixed state, when in fact, it is zero which I corrected myself to in my second comment.