question about eladrin presences

[ATHATH]

Autumn and Spring Eladrins, from Mordenkainen's Tome of Foes, have the following ability:
Enchanting Presence. Any non-eladrin creature that starts its turn within 60 feet of the eladrin must make a DC 16 Wisdom saving throw. On a failed save, the creature is charmed by the eladrin for 1 minute. On a successful save, the creature becomes immune to any eladrin's Enchanting Presence for 24 hours.
Whenever the eladrin deals damage to the charmed creature, the creature can repeat the saving throw, ending the effect on itself on a success.

If a creature has been charmed by this Enchanting Presence, does it still have to make another saving throw against it each time it starts its turn within it? If so, if it succeeds on one of those saving throws, does it immediately break free from its enchantment/charmed state due to now being immune to it for 24 hours?

[Aett_Thorn]

Autumn and Spring Eladrins, from Mordenkainen's Tome of Foes, have the following ability:
Enchanting Presence. Any non-eladrin creature that starts its turn within 60 feet of the eladrin must make a DC 16 Wisdom saving throw. On a failed save, the creature is charmed by the eladrin for 1 minute. On a successful save, the creature becomes immune to any eladrin's Enchanting Presence for 24 hours.
Whenever the eladrin deals damage to the charmed creature, the creature can repeat the saving throw, ending the effect on itself on a success.

If a creature has been charmed by this Enchanting Presence, does it still have to make another saving throw against it each time it starts its turn within it? If so, if it succeeds on one of those saving throws, does it immediately break free from its enchantment/charmed state due to now being immune to it for 24 hours?

It doesn’t say that they get a save at the beginning or end of their turn, so they don’t. They only get a new save if the Eladrin does damage to them, from the text.

[Millstone85]

It doesn’t say that they get a save at the beginning or end of their turn, so they don’t. They only get a new save if the Eladrin does damage to them, from the text.The question isn't about whether they get saves by default, but about what happens if they start their turn within the eladrin's enchanting aura while already affected by it.

My guess is that:

They do make another saving throw each time they start their turn in the aura.
A successful saving throw doesn't immediately break the charmed state. It just means they won't have to make another save against this aura for 24 hours.