Is there a rigorous upper bound on phonon-mediated $T_{\mathrm{c}}$
In plain words
For superconductors glued by lattice vibrations, the transition temperature seems to stay below about one tenth of the vibration energy scale, and the vibration energies themselves are capped by fundamental constants; neither statement is proven.
Precise statement
Within Migdal-Eliashberg theory and beyond it, determine whether $k_B T_c \le A \hbar \omega_{\mathrm{bar}}$ holds with a universal constant A ($A \sim 0.1$ argued from the breakdown of Migdal theory at strong coupling), and combine it with bounds on the highest phonon frequency in terms of the electron mass, charge and Planck constant to give a material-independent $T_c$ ceiling. An answer is a theorem with stated assumptions, or a model counterexample with controlled $T_c$ above the bound.
What would settle it
A proof within a controlled model of electrons coupled to phonons (for example Holstein at all couplings via exact numerics) or an explicit counterexample.
Status in the literature
Unverified note
Bounds proposed in 2018 and 2025 are heuristic or order-of-magnitude, placing the ceiling at about $10^{2}$ to $10^{3}\,\mathrm{K}$.