Can You Solve this Ice Diamond Riddle? ft. Vsauce’s Michael Stevens

It goes DOWN. If the ice cube floats despite containing the diamond (which seems implied by the wording and image), then the diamond displaces a volume of water the same weight as its own when floating but only the same volume as its own when it sinks. If OTOH the cube + diamond sink immediately then the diamond always displaces the same volume of water as its own but it is now the ice that takes more volume when solid than when melt.