Edit: I had a simply incorrect attempt to show that $\mathfrak{m}_T$ is closed under taking absolute values.

This is equivalent to showing that $\mathfrak{m}_T$ is closed under absolute values. If $a \in \mathfrak{m}_T$ is self-adjoint, then again by the continuous functional calculus, $|a| = a_+ + a_-$, with $0 \leq a_+ \leq |a|$ and $0 \leq a_- \leq |a|$.