Let $A$ be a densely defined symmetric (unclosed) operator and let $B\in B(H)$ be positive.

I know that if $\overline{A}B$ is normal, then $AB$ need not be normal.

My question is: If $AB$ is normal, is it necessary that  $\overline{A}B$ remains normal? (notice that, thanks to the normality of $AB$, I have shown that $B\overline{A}\subset \overline{A}B$).

Many thanks,

Hichem