Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, rain starts. As a result,

The buckets slow down because the angular momentum of the bucket + rain system is conserved.

The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved.

The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane.

The buckets speed up because the potential energy of the rain is transformed into kinetic energy.