Use that regardless whether she receives extension letter or not this week.
The benefits of this approach is:
- Less confusion by HR
- No need to do reverification
When LPR presents GC and extension letter many HRs get confused, not knowing whether one is still authorized to work or not. Also, they keep asking for updates on I-751 status and ask for new extension when current one is expiring. Best to not to go this route altogether.
The only places that need to see GC and extension letter are:
- Airlines / CBP when going to the US
- DMV to renew Drivers License
- Banks when confirming residency status (this is debatable, but some ask when applying for loans, mortgage).
- Immigration lawyer if representing LPR
- Police if confirming residency status
Everybody else should be sarisfied with DL + SS card, including employers.