When I've seen/done Boc protections with water as a solvent it usually has an excess of sodium bicarb or similar around, though ostensibly it isn't really necessary. Probably isn't necessary for them to be running it under argon either
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I wouldn't worry about the solvent choice, its a relatively common mix for various reaction sets.
Neutralizing with conc. HCl has to be carefully monitored. They do say they bring it down to 2.9 which is perfectly fine, especially for their conditions at the time (chilled with an organic layer on top).
I'd say around pH=1 you will start to get deprotection at rt. I do most of my boc deprotections with TFA/DCM/H2O mixes, and that usually takes a few hours depending on substrate.