The materials and body were clean. The problem was very strange because the specks only appeared on the insides of the ware. The problem turned out to be iron powder in the burners (shown in the overlay in this photo). Disassembling and cleaning them solved most of the issue. The rest? Disassembling and cleaning them better!
Specking is a fault in fired ceramic ware where speckles of ion bearing contaminants are visible and mar the appearance of the surface.