I searched and found some similar issues, but not exactly this one...
When I delete my messages from my ISP (using Verizon web mail), the messages remain on the BB. I have tried activating the checkbox on the BIS website to sync deleted items, and this has not changed a thing.
When I delete on the device, the messages delete from the host server very quickly, but not vice versa. I am forced to manually delete on the device every time I delete from the ISP site.