Maximal volume conjecture for truncated tetrahedra with equal edge lower bounds

Let ${\mathcal{T}}_\ell$ denote the space of truncated hyperbolic tetrahedra whose edge lengths satisfy the constraint defining ${\mathcal{T}}_\ell$, and let $\Delta_\ell$ be the regular truncated tetrahedron in this space. Maximal volume conjecture. For every positive real number $\ell$ and every $\Delta\in{\mathcal{T}}_\ell$,

$$\operatorname{vol}(\Delta)\leq\operatorname{vol}(\Delta_\ell),$$

and equality holds if and only if $\Delta=\Delta_\ell$. The theorem proved in the paper establishes this only under the technical assumption $\ell\leq\ell_0$; the unrestricted assertion is expected to imply the corresponding result for the ideal simplicial volume of hyperbolic $3$-manifolds with geodesic boundary, but remains open in general.