umount2() for an NFS mountpoint returns EPERM, even when called as root

I'm running Ubuntu 20.04, but I've had this problem back to Ubuntu 14.04.

I've been trying to automate unmounting NFS mounts when my VPN connection closes, but no matter how I call "umount", I get "Operation not permitted" even though I'm running with root privileges. I thought it would be easier to diagnose the issue if I just wrote a small stand-alone program to do the unmounting:

int main(int argc, char **argv)
{
 int rc; 

   rc = setuid(0); 
   printf("setuid: rc %i, %s\n", rc, strerror(errno));

   rc = umount2(argv[1], MNT_FORCE | MNT_DETACH); 
   printf("umount2: rc %i, %s\n", rc, strerror(errno));

   return 0;
}

When I run this in a shell session, I get exactly what I expect:

setuid: rc 0, Success
umount2: rc 0, Success

When I run it in a script triggered by my VPN software (OpenVPN), I get this:

setuid: rc 0, Success
umount2: rc -1, Operation not permitted

The setuid() call works, so I know I'm running as root (and in fact, the program is called already running as root, so this shouldn't be required anyway).

What other permission would umount2() be looking for that's not just UID 0?