"A +1 short sword with a thin, dull gray blade, this weapon provides a +4 bonus on its wielder’s attack and damage rolls when she makes a sneak attack with it."
So +4 to the weapon damage roll. **NOT** +4 to the sneak attack roll, and certainly not +4 to EACH DIE of the sneak attack roll.
Or am I wrong?
I would assume that it is +4 to the total sneak attack bonus damage as immunity to sneak would provide immunity to the bonus, but perhaps that isn't strictly RAW by the wording.
Maybe it was meant to be applied to the sneak attack damage, but the current wording only says "damage" which is your base weapon damage. If it was meant to apply to sneak attack, then it should say something like "sneak attack damage" or "damage of your sneak attack".
It most certainly does not apply per die in any circumstance. If it applied per die it would have stated that specifically.
The only thing that this will make a difference on is if it applies to the sneak attack roll it would be considered precision damage. If that is the case the damage would not be multiplied on a critical hit. Since the description does not mention it being precision damage and states it applies to the damage rolls it should be applied the normal damage roll and be multiplied on a critical hit.
Another clue it the fact that you apply the bonus to the attack roll. Sneak attack is simply a bonus to damage and does not have an attack roll. A sneak attack uses the attack roll of the attack it is being added to. If the bonus only applies to the sneak attack, it does not increase your chance of hitting which the ability obviously does.