The Chestnut system is designed for the Mandrels to be held in a self centring chuck, (18-25 or 10mm registers) not on a Morse taper.
Be aware that any system that uses a Morse taper to drive the mops should have either a tailstock safety centre (prevents internal bowl mop use) or a headstock drawbar facility to prevent the taper coming out of the headstock during use if vibration breaks the taper adhesion.
If a taper lets go in best case scenario the mop will stop rotating and just fall out, in worst case it will spin wildly on a rapidly developing cone/arc with the centrifugal force and throw the mop and mandrel as far as the nearest obstacle.