Let $T\in B(H)$. Assume that $T^3$ is normal, i.e., $T^{*3}T^3=T^3T^{*3}$. When is $T^2$ normal?

Here is a known related result:

Also, if $T^2$ is normal and $T$ has a positive real part, then $T$ is normal.

Feel free to share any ideas or insights.