Fong–Tsui conjecture for bounded operators

Let $T$ be a bounded operator on a complex Hilbert space. Write

$$\operatorname{Re} T=\frac{T+T^*}{2}$$

for its real part, and let $|T|$ denote the positive square root of $T^*T$.

Fong–Tsui conjecture. If

$$|T|\leq |\operatorname{Re} T|,$$

then $T$ is self-adjoint.

This conjecture asks whether domination of the absolute value of a bounded operator by the absolute value of its real part forces self-adjointness. It was proposed as a strengthening of an earlier result of Fong and Istrătescu; the supplied source treats it as an open question.