It depends on whether you think "Whenever one takes damage" means "Whenever one takes damage for each source of damage" or "Whenever one takes damage for each distinct time damage occurs".
There really hasn't been clear guidance given that I can find, and really this ends up coming down to the same question as "Would striking someone at 0 hit points with an entire Magic Missile spell automatically kill them?" and the analysis in my answer there. The only precedent we have is that Concentration checks specifically check for each source of damage (as "If you take damage from multiple sources" is explicitly called out in the Concentration rules), but other rules that look for damage (such as the one you ask about) don't make clear if they also trigger for each source as well (as a Magic Missile has three sources) or whether they just trigger for each time (as a Magic Missile's damage explicitly happens simultaneously).
It's really about whether you look on the concentration rule as being different from the "normal" case (since it calls out sources explicitly and other rules do not) or whether you consider it to be the normal case as how to handle damage triggers is spelled out more specifically there and it should be used as a precedent for places that it's not spelled out.
As with all the rules, reading and interpreting them is up to your DM. If you're the DM, just go with whichever interpretation you think is most correct or will lead to the most fun gameplay, and ensure your players know how your game will work. At the end of that day, while it might be nice if the rules answered clearly all our questions like this one, the rules are just guidelines for having fun with your group, and they're there to serve you and not the other way around.