You would be able to breathe. The effect of the necklace of adaptation
wraps the wearer in a shell of fresh air, making him immune to all harmful vapors and gases […] and allowing him to breathe
When it continues to say “even underwater or in a vacuum”, that's what's known as an “elaboration”, a normal part of how English gets used. It doesn't restrict, it clarifies; the “even” as used there is synonymous with “including” in the non-exhaustive sense.
In order to be a restriction it would have to omit the comma and the “even”, which is a very different sentence construction.
In other words, the necklace's effect doesn't care about the environment at all, allowing normal breathing regardless of environment. (Even underwater or in vacuum!)