Does downing a character at the start of its turn require an immediate Death Saving Throw? [duplicate]

This question already has an answer here:

  • When does the Death Saving Throw happen with respect to sources of automatic ongoing damage? [duplicate] 1 answer

Say a Playable Character (PC) with low HP is in a Cloudkill:

When a creature enters the spell’s area for the first time on a turn or starts its turn there, that creature must make a Constitution saving throw. The creature takes 5d8 poison damage on a failed save, or half as much damage on a successful one.

The PC is next in initiative and starts its turn. The only effect to resolve is the Cloudkill. It fails the save and the resulting damage is sufficient to get him to 0 HP.

Since this event occurred “at the start of the turn” and this is also when Death Saving Throws happen for a downed character, does the PC need to make an immediate Death Saving Throw? Or did has the “start of the turn” passed and the first Death Saving Throw will only start next round?

I am aware of this relevant question “How to resolve multiple ‘start of turn’ triggers” and the accepted answer: events occurring at start of turn are resolved in the order chosen by the player.

However, since in my example the events cannot be reversed (the PC cannot make a Death Saving Throw before taking the damage), I feel that the question stands on its own.