Do superfluid-stiffness bounds on $T_c$ extend to three dimensions
In plain words
In two-dimensional superconductors the transition temperature is provably limited by how stiff the superconducting phase is, which in turn is limited by how easily electrons move; whether a similar proof works in three dimensions is unknown.
Precise statement
For two-dimensional superconductors the Berezinskii-Kosterlitz-Thouless relation and the optical sum rule give rigorous upper bounds on $T_c$ in terms of band dispersion and density (2019). Determine whether a comparable rigorous bound on $T_c$ exists in three dimensions, and in flat-band systems where the superfluid stiffness has a quantum-geometric contribution. An answer is a proof with stated assumptions or a counterexample model.
What would settle it
A proof bounding $T_{c}$ by stiffness or kinetic-energy sum rules in 3D, or a solvable 3D model violating any such bound.