Correct. This can cause significant headaches for green card holders; confiscation of their physical green card and detention. Here's Justice Jackson explaining the broad theory and the way it applied in Lau's case in particular ("LPR" here means "Lawful Permanent Resident" aka green card holder).
Aaron Reichlin-Melnick
But it does mean that border officials can just declare, without evidence, that green card holders are applicants for admission, which is both crazy and makes things deeply difficult for green card holders.
