Sorry wasn't able to get back to you sooner.
The area is east of US 54 and roughly east of the Oro Grande area and just north of the Texas border. Some of it is under control of the US Army (NM segment of Fort Bliss) and the rest under BLM management. The eastern boundary is the Guadalupe Mountains. NM non-military access to it is via a road eastward at the Highway 54 Border Patrol checkpoint.
But, even if it's on the east side of 54, instead of the west, that doesn't change the "empty quarter" description, does it?