I just played pretend bookings and Turkish well & truly undercut LH with Air China and a Middle Eastern carrier also below LH. However these were for the most basic of fares. But for more flexibility and greater comfort the price range between carriers evens out allowing SQ & TG to become competitive.
For praa's chosen route for TG & SQ to offer the competitive fares he seeks would involve undercutting their own Munich > home base fares for an extended journey through their own hubs. As these companies must know the market on their own spokes this surely answers praa's question.