When I find a seated core sticking to my punch when the die is lowered, its because I haven't set the punch deep enough.
When experimenting with the die/punch combo, I lube after each attempt that the seated core remains on the punch. When the core is seated with enough pressure, the expanded jacket clutches the sides of the die and remains inside when the die is moved away from the punch. Its then ejected from the die in th e normal manner.
I haven't had any experience with tapered jackets, but I'd expect that you should be able to determine whether any of your jackets are tapered quite easily. If you can't get a caliper inside the jacket mouth, you could make a casting or something from a moderately maleable material.