The link between SK and Lockheed is pretty strong with the KAI T-50, which will be rebadged as an LM product for USAF T-X competition. Suspect there is a lot of high level sales effort going in, but as you say it's the 'recommended' solution.
If Singapore buys F-35 as well then politically could SK be seen as a leading regional power without it? China will have the J-31 eventually so SK looks to be a poor relation without a 'stealth' jet. That is a whopping price tag so I suspect it'll come down to the size of their defence budget.