Guts determine the leading coefficients of $L^2$-Alexander torsions

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.