Heritability of the property $\mathcal D_\pi$ by overgroups of $\pi$-Hall subgroups in the case where $2\in\pi$

Let $\pi$ be a set of prime numbers. We say that a finite group $G$ is a $\mathcal D_\pi$-group if all of its maximal $\pi$-subgroups are conjugate. Question 17.44(b) in Unsolved Problems in Group Theory, The Kourovka Notebook, asks whether an overgroup of a $\pi$-Hall subgroup of a $\mathcal D_\pi$-group is always a $\mathcal D_\pi$-group. We give an affirmative answer to this question in the case where $2\in\pi$.