Casimir repulsion between solid bodies separated by vacuum
In plain words
Casimir forces are almost always attractive, and they have been made repulsive only with a liquid between the objects. Predicted repulsion across empty space, using magnetic, chiral or topological materials, has never been seen.
Precise statement
Repulsive Casimir-Lifshitz force between two solid bodies across vacuum. Mirror-symmetric arrangements of identical bodies always attract (Kenneth-Klich theorem, 2006); repulsion is predicted for dielectric-magnetic pairs (Boyer, 1974), topological insulators with axion electrodynamics, and chiral metamaterials, but material dispersion usually restores attraction. An answer is a measured repulsive force with sign and magnitude matching Lifshitz theory for the measured permittivities and permeabilities, or a theorem that excludes repulsion for all passive materials at accessible gaps.
What would settle it
A sphere-plate or plate-plate force measurement across vacuum showing a repulsive force at a given gap, reproduced by Lifshitz theory with independently measured material response.
Status in the literature
Repulsion has been measured only across fluids (Munday, Capasso and Parsegian, Nature 2009).