Maximum $T_c$ from purely repulsive electron interactions
In plain words
If electron repulsion alone can make electrons pair, there should be a largest possible transition temperature, measured in units of the energy for an electron to hop between neighboring atoms; in cuprates the transition temperature is only about 3 percent of that hopping energy.
Precise statement
For lattice models with purely repulsive interactions (Hubbard-type, bandwidth $W=8t$ on the square lattice), determine the maximum $T_c/t$ over interaction strength, filling and lattice geometry, and whether it exceeds the cuprate value $T_c/t\sim 0.03$. An answer is controlled numerical $T_c$ values with error bars, or a bound.
What would settle it
Unbiased finite-temperature calculations (DQMC, diagrammatic Monte Carlo, cluster methods with size extrapolation) of $T_c$ across parameter space.