Does Calm Emotions prevent the damage from Weird?

Weird says

On a failed save, a creature becomes frightened for the duration. The illusion calls on the creature’s deepest fears, manifesting its worst nightmares as an implacable threat. At the end of each of the frightened creature’s turns, it must succeed on a Wisdom saving throw or take 4d10 psychic damage. On a successful save, the spell ends for that creature.

Calm Emotions Says

You can suppress any effect causing a target to be charmed or frightened. When this spell ends, any suppressed effect resumes, provided that its duration has not expired in the meantime.

From this question, we’ve determined that calm emotions does not prevent saving throws against fear effects. This implies you would still make the saving throw against Weird. Does that mean that despite suppressing the effects of Weird, you might still take damage?

I’d like to know how people interpret the rules here, but also how you might rule it at your table. I’m inclined to think it’s silly to still take damage despite suppressing the fear, and would probably rule that you make the saving throw to end the effects of Weird, but if you fail you do not take damage.