A major challenge in fluid dynamics is assessing the nonlinear stability of laminar flows under arbitrary perturbations. The energy method, introduced by Reynolds and Orr, is the main technique used to prove this. Joseph and Carmi explored this problem in annular and circular domains in their acclaimed 1969 paper, where, in particular, they reported the energy stability limit for Hagen-Poiseuille (pipe) flow and 3D (doubly periodic) plane Poiseuille flow. In annular domains, defining η as the ratio of the inner and outer radii, they found that usually, for modest values of η, the most energy unstable eigenmode was inherently two-dimensional and independent of the streamwise direction. However, they also found that in a pipe, the most energy unstable mode was three-dimensional, corresponding to a spiral mode with the simplest nontrivial azimuthal periodicity. Despite the fact that an annular domain represents a fundamental topological modification, they conjectured that for a sufficiently small inner radius (η<10-4) the most energy unstable mode would transition to become three-dimensional. Nevertheless, they were unable to compute such a case, nor were they able to prove it. In this work, we employ a mixed variational formulation in primal variables (i.e. three components of velocity and pressure) for the energy eigenproblem and, assuming a periodic streamwise wavenumber α≥0, we discretize it using a finite element method. Exploiting its flexibility in terms of reproducing the geometry, we were able to computationally confirm Joseph and Carmi’s 55-year-old conjecture for the first time, explicitly finding the value of η where this transition occurs. Furthermore, Joseph and Carmi’s proved that energy stability eigenvalue branches, as a function of α, always exhibit a local minimum at α=0. By exploiting the functional analytic framework of our variational formulation, we were able to generalize that theoretical result to ducts of arbitrary cross-section.