It seems that this is ultimately the problem of inhibiting the creation of shadow rays from camera rays incident on the floor. Therefore, we can detect them by looking for rays with a 'Ray Depth' equal to 1. Since we only care about rays coming from the floor, we can use a separate object positioned above it to act as a filter.
Ideally, this would be it: use a mix shader to permit everything but shadow rays with depth equal to 1 from passing through. Unfortunately, it seems that the glossy shader labels some of its shadow rays as glossy, so until this is fixed you have to work around it by selectively making parts of the floor purely diffusive.
Using the naive solution:
Example using a somewhat smooth, metallic 'Principled BSDF' in place of 'Glossy BSDF'.