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Do the Lee sizing dies in .40 have a different taper at the mouth than Hornady carbide dies and would the Lee die allow the cases to enter the die more easily?
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Lee dies are no better, as far as the chamfer/fillet at the mouth of the die.
Some RCBS dies do have a wider mouth, but the chamfer is a shallower angle. So, it doesn't help much. (And, I don't know about .40 S&W.)
Dillon dies are substantially better, but I don't know how well they work on the LNL AP.
If you're having problems with case alignment
and primer seating, I would take a close look at indexing. I don't have much experience on the LNL AP, so I can't be more specific.