When you convert an offhand with Jasper will it remain the same offhand or will it change too a class specific one?

It will change to a class specific one.
This will be the pattern for it.
Taken from the DQ Wiki just go to the bottom page or search Jasper

Item Conversions (Jasper)
Wizard Warrior Rogue
Staff Sword Chakram
Wand Lance Flintlock
Sword Hammer Dagger
Gauntlet Axe Bow

Wizard Warrior Rogue
Manashield Hatchet Mirror
Skull Horn Trap
Tome Totem Vial
Orb Shield Bomb

Thanks : D darn though wanted a skull on my pvp rogue but that’d be pretty op though with my build so understandable