Dihedral-angle averaging conjecture for truncated tetrahedra

Let $\ell\in\mathbb{R}$, let $\Delta\in{\mathcal{T}}_{\ell}$, and let $\theta_{ij}$, $1\leq i<j\leq4$, be the dihedral angles of $\Delta$. Set

$$\theta'=\frac{\theta_{12}+\theta_{13}+\theta_{14}+\theta_{23}+\theta_{24}+\theta_{34}}{6},$$

and let $\Delta'\in{\mathcal{T}}$ be the regular tetrahedron whose six dihedral angles all equal $\theta'$. Dihedral-angle averaging conjecture. Then $\Delta'\in{\mathcal{T}}_{\ell}$. Equivalently, if every edge length of $\Delta$ is at least $\ell$, the regular tetrahedron obtained by assigning the average dihedral angle of $\Delta$ to every edge has edge length at least $\ell$. This conjecture would imply the maximal-volume conjecture above; it is presented as open in the source.