Self-adjointness conjecture for operators with positive real part and self-adjoint powers

Let $H$ be a Hilbert space and let $T\in B(H)$, where $B(H)$ denotes the bounded operators on $H$. Suppose that the real part of $T$ is positive and that $T^3$ and $T^4$ are self-adjoint.

Self-adjointness conjecture. $T$ is self-adjoint.

This conjecture asks for a generalization of the preceding theorem, which establishes self-adjointness under positivity of the real part together with self-adjointness of $T^2$ and $T^3$. The status of the proposed $(3,4)$-power generalization is not specified in the source.