dnd 5e – What damage type is empowered when Ice Storm is cast by an evoker?

The party Evoker has just reached 10th level and gained Empowered Evocation

Beginning at 10th level, you can add your Intelligence modifier to one damage roll of any wizard evocation spell you cast.

While most of his evocation spells do a single type of damage, he does have access to Ice Storm, which does

2d8 bludgeoning damage and 4d6 cold damage on a failed save, or half as much damage on a successful one.

It could matter which type of damage has been empowered, for example when the target has a different resistance, immunity, or vulnerability to bludgeoning vis a vis cold.

Which type, or types, of damage are increased when Ice Storm is cast with empowered evocation?

There are related questions already on this site addressing, when there are multiple damage types, which damage type…

the Sharpshooter feat applies to ranged damage

the Sneak Attack class feature applies to damage

damage-reducing effects like the class feature Cutting Words reduces

The consensus from the answers to these questions is that even though the damage itself counts as “one roll”, it does have components of different types of damage, and that damage increases are applied to one type, only. These answers also affirm that there is no RAW for which damage type is increased, and as such mostly advocate for allowing the player using the ability to choose.

However, in the case of my question, I am leaning towards saying that the damage increase applies specifically to the cold damage, because flavor-wise cold feels more like what would be enhanced by evocation magic (the bludgeoning is more like conjuration, I think). A good answer will either support or refute my opinion on this.

Note this question is about whether empowered evocation applies to just one of the damage components of Ice Storm, or both. In their answer, wax eagle says “which damage type it falls under is not addressed, and as such I would allow the wizard to pick. If you want to do something consistent, always applying it to the first damage instance would make sense”. This is a possible answer to my question, although I would prefer that it address my assumption that it is more natural for the ability to apply to the cold damage.

Finally, for what it is worth, this is the same evoker PC as is referenced in this question.