Amadeus doesn't have a reservations system partner in the US so winning Southwest would be a major win.
Amadeus signed a contract several years ago to build a reservation system for united airlines but implementation never happened,although the contract remains in place United Airlines backed off out of financial concerns
Sabre has been making inroads with its sabresonic reservations system in north america with implementations with jetblue and westjet,although Sabre lost out to HP,which picked up a contract to create a new reservations system for american airlines.
Southwest is the largest domestic carrier in us and it's reservations system contract would be a prize for sabre or amadeus