If $A(z) :=[A_{ij}(z)] $ and $B(z) :=[B_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A_{ij}(z)$, and $B_{ij}(z) $ with

(1). $AA^{\#}=A^{\#}A$ where $A^{\#}(z)=\left(\overline{A(\bar{z})} \right)^{T}$.

(2). $BB^{\#}=B^{\#}B$

(3).$ AB=BA$

Is it true that $$B(A A^{\#}) =(AA^{\#}) B$$.

Thanks in advance.