Symmetric-orbit convex-hull conjecture for truncated tetrahedra

Let ${\mathcal{T}}$ be the space of marked truncated tetrahedra, let $\Theta:{\mathcal{T}}\to{\mathcal{O}}$ record their dihedral angles, and let the symmetric group $\mathfrak{S}_4$ act by permuting the four removed vertices. For $\sigma\in\mathfrak{S}_4$, write $\sigma\cdot\Delta$ for the corresponding symmetric image. Let $\ell\in\mathbb{R}$ and let $\Delta\in{\mathcal{T}}_\ell$ be non-regular. Define ${\mathcal{O}}_\Delta\subseteq{\mathcal{O}}$ as the convex hull of

$$\{\Theta(\sigma\cdot\Delta):\sigma\in\mathfrak{S}_4\}.$$

Symmetric-orbit convex-hull conjecture. The intersection

$$\left({\mathcal{O}}_\Delta\setminus\{\Theta(\sigma\cdot\Delta):\sigma\in\mathfrak{S}_4\}\right)\cap\Theta({\mathcal{T}}_\ell)$$

is non-empty. Thus the convex hull, with respect to dihedral angles, of the symmetric images of any non-regular $\Delta$ contains another point of ${\mathcal{T}}_\ell$ besides its orbit points. The source presents this as a related open conjecture, motivated by symmetry and convexity methods for proving the regular tetrahedron's volume-maximizing property.