Data point
I don't work for SWA. I wrote an external LOR for a former colleague, and received a phone call from SWA hiring after his interview to follow up / verify what I had written. Sounded like they were compiling the interview notes and application package for final review.
It was a two min conversation, half of which was "how are you liking American?", though I think it would have been longer if the letter was less substantive.
FWIW...