On a conjecture of $\lambda$-Aluthge transforms and Hilbert--Schmidt self-commutators

Let $A$ be a complex square matrix, and write its polar decomposition as $A=U\mathrm{\mid }A\mathrm{\mid }$. For $0<\lambda <1$, the $\lambda$-Aluthge transform of $A$ is defined by $${\mathrm{\Delta }}_{\lambda }(A)=\mathrm{\mid }A{\mathrm{\mid }}^{\lambda }U\mathrm{\mid }A{\mathrm{\mid }}^{1-\lambda }\mathrm{.}$$ In 2007, Huang and Tam conjectured that the Frobenius norm of the self-commutator is contractive under ${\mathrm{\Delta }}_{\lambda }$: for every $0<\lambda <1$, $$\mathrm{\parallel }{A}^{\ast }A-A{A}^{\ast }{\mathrm{\parallel }}_F\ge \mathrm{\parallel }{\mathrm{\Delta }}_{\lambda }(A{)}^{\ast }{\mathrm{\Delta }}_{\lambda }(A)-{\mathrm{\Delta }}_{\lambda }(A){\mathrm{\Delta }}_{\lambda }(A{)}^{\ast }{\mathrm{\parallel }}_F\mathrm{.}$$ If this inequality held, then the iterated self-commutator norms $$\{\parallel {\mathrm{\Delta }}_{\lambda }^m(A{)}^{\ast }{\mathrm{\Delta }}_{\lambda }^m(A)-{\mathrm{\Delta }}_{\lambda }^m(A){\mathrm{\Delta }}_{\lambda }^m(A{)}^{\ast }{\parallel }_F{\}}_{m\in \mathbb{N}}$$ would form a nonincreasing sequence and necessarily converge to $0$. In this paper we provide a counterexample, thereby disproving the conjecture. We also obtain the quantitative bounds $$\sqrt{\frac{3}{2}}\le \underset{\begin{array}{c}A\in {\mathbb{M}}_n(\mathbb{C}),A^{\ast }A\mathrm{\ne }A{A}^{\ast }\\ 0<\lambda <1\end{array}}{\sup }\frac{\mathrm{\parallel }{\mathrm{\Delta }}_{\lambda }(A{)}^{\ast }{\mathrm{\Delta }}_{\lambda }(A)-{\mathrm{\Delta }}_{\lambda }(A){\mathrm{\Delta }}_{\lambda }(A{)}^{\ast }{\mathrm{\parallel }}_F}{\mathrm{\parallel }{A}^{\ast }A-A{A}^{\ast }{\mathrm{\parallel }}_F}\le 2.$$