Arrows of Decay being shot from a bow enchanted with Flame don't give wither damage

Arrows of decay being shot from a bow enchanted with Flame give a wither effect for 5 seconds, but since the Arrow of Decay is setting the mob or player on fire, the fire damage seems to overwrite the Wither damage.
If the entity is in water, rain, or under the effect of fire resistance, the wither effect will damage it. Blazes get the Wither damage too.
When shot with a non-flame bow, the damage is taken as normal.