On Fuglede-Putnam type theorems for generalized *-Aluthge transform

Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces and let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A\in \mathbb{B}(\mathcal{H})$ and $B\in \mathbb{B}(\mathcal{K})$ and let ${\rm Com}(A,B)$ stand for the set of operators $X\in \mathbb{B}(\mathcal{K},\mathcal{H})$ such that $AX=XB$. Let $\tilde{C}^{(*)}_{(s,t)}$ denote the generalized $*$-Aluthge transform of a bounded linear operator $C$. We investigate that under what conditions on operators ${\rm Com}(A,B)={\rm Com}(\tilde{A}^{(*)}_{(s,t)},\tilde{B}^{(*)}_{(s,t)})$. Also we prove an inequality for Schatten $p$-norms related to this equality.