That's true at a high level, but off on some dates.
Southwest cut a deal with Amadeus in 2007, but is still using the internal homegrown copy of Sabre for all domestic bookings. If you book something that lands outside the US, it goes via Amadeus. They will move all reservations to Amadeus, but over the next year or so.
They exposed bookings to corporate booking tools starting in 2009, but not via a GDS, but rather by a private connection.[1]
Then, in 2015, they exposed bookings via Amadeus, including domestic, to corporate travel tools. But, the booking itself does land in the aforementioned homegrown reservation system.
[1]https://www.concur.com/newsroom/article/concur-and-southwest...