There are so many factors that can play a role here that a "one size fits all" answer is impossible.
Everybody's setup and equipment origin/history is a little different.
You can mount the headstock end of the mandrel in a MT taper, in a chuck's jaws, in a collet chuck, etc. . All will work and work well depending on "other things" which only the operator can certify as OK.
At the tailstock end, the quality and fit of your live center, and the fit of your MT taper to the tailstock body (including the possibility of foreign matter) are factors. Also the shape of the point of your live center and the shape of the dimple in end of mandrel, as others have said. . Not to mention the bearings in your live center ... they have a limited life and are often a reason to discard a live center or give it to a starting-out pen maker !! :biggrin:
Even with a mandrel saver you have worries like foreign matter inside the tailstock body, clearance fit of the mandrel to the mandrel saver, bearings in the mandrel saver, inaccuracies on the tapered arbor of the mandrel saver, etc. ... even the tightness of the locking lever on your tailstock can play a role.