calculus and analysis – What results of FunctionMeromorphic are true?

Studying a new command of 12.2 FunctionMeromorphic, I try

FunctionMeromorphic((Re(z) - I*Im(z))/Abs(z)^2, z)

False

FunctionMeromorphic((Re(z) - I*Im(z))/Abs(z)^2 // Simplify, z)

False

FunctionMeromorphic(((Re(z) - I*Im(z))/
Abs(z)^2 /. {Re(z) -> (z + Conjugate(z))/2, 
Im(z) -> (z - Conjugate(z))/2, 
Abs(z) -> Sqrt(((z + Conjugate(z))/2)^2 + ((z - Conjugate(z))/2)^2)}) //FullSimplify, z)

False

FunctionMeromorphic((Re(z) - I*Im(z))/Abs(z)^2 // FullSimplify, z)

True

I suspect the latest result is not true. I am right, aren’t I?