My initial reaction to this question was, "it shouldn't" but I did a little research and I think this question has a lot of legitimacy.

As far as I can tell, if you activate the SOS feature on an InReach it sends an SMS message to the Rescue Coordination Center (RCC) privately operated by GEOS...with a link to your GPS coordinates. This gives me the impression that if GEOS was using AT&T for their SMS connectivity, the same issue could have occurred.

Note that I have no association with Garmin, GEOS, or AT&T other than as a customer, and I have no inside information on these matters.