  For 3-manifolds, the leading coefficient of the $L^2$-Alexander torsion is a
numerical invariant of a real first cohomology class. We show that the leading
coefficient equals the relative $L^2$-torsion of the manifold cut up along a
norm-minimizing surface dual to the cohomology class. Furthermore, the leading
coefficient equals the relative $L^2$-torsion of the guts associated to the
cohomology class. Finally, we prove that the leading coefficient is constant on
any open Thurston cone. The main ingredients are a new criterion for the
convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on
guts of 3-manifolds.
