logic – validity of proof (100 green eyes riddle)

Here is the 100 green eyes riddle:

Imagine an island where 100 people,all perfect logicians,are imprisoned by a mad dictator.There’s no escape,except for one strange rule. Any prisoner can approach the guards at night and ask to leave. If they have green eyes, they’ll be released. If not, they’ll be tossed into the volcano. As it happens, all 100 prisoners have green eyes, but they’ve lived there since birth, and the dictator has ensured they can’t learn their own eye color. There are no reflective surfaces, all water is in opaque containers, and most importantly, they’re not allowed to communicate among themselves. Though they do see each other during each morning’s head count. Nevertheless, they all know no one would
ever risk trying to leave without absolute certainty of success.

On a morning, they are provided the information that “At least one of you has green eyes.”
causing that on the hundredth morning after the sentence all the prisoners are gone.

The riddle and solution were available here:
https://www.youtube.com/watch?v=98TQv5IAtY8

In their proof, they consider that having the information that “At least one of you has green eyes.”in case of 2 green-eyes people causes them to leave on the second day. This same element being used to prove the same conclusion for 3 green-eyed people on the third day etc.. which leads to conclusion that on the 100th day, all the 100 green eyed people have left.

My question is:

Is it valid that a group of 2 green eyed people would escape on the second day, since actually the sentence “At least one of you has green eyes.”, was only pronounced in front of the group of 100 ?