Margulis constant of orientable hyperbolic 3-manifolds

Prove that the three-dimensional Margulis constant is equal to the optimal Margulis number of the Weeks manifold m003(-3,1), namely that \(\mu_3=\mu(\mathrm{m003}(-3,1))=0.77442660700998\dots\).

Background

The paper computes verified optimal Margulis numbers for the orientable cusped census up to 11 tetrahedra and for the closed SnapPy census, except for m007(3,1). Among the manifolds considered, the Weeks manifold m003(-3,1) has the smallest known optimal Margulis number, providing the upper bound $\mu_3\leq 0.77442660700998\dots$.

The conjecture asserts that this census-derived upper bound is sharp globally over all orientable, finite-volume hyperbolic 3-manifolds. Establishing it would determine the exact value of the three-dimensional Margulis constant.