While I do find it odd that Hyundai wouldn't recommend synthetic for a turbo/di motor, I'm not engineer so don't have ground to dispute what they're saying. With that being said, I would use any conventional that meets the specifications recommended by the manufacturer.
As a side bar- It could be that manufacturers know most "conventional" oils today do in fact have some Group III and above base oil in the formulation. Due to this it is likely that the conventional oils perform much better than what they're tested for (in the eyes of the manufacturers), this, added with the cheaper cost-to-own conventional will provide that's been pushed my the rags and accountants could be the reason they do not recommend/require syn. At the end of the day, an auto manufacturer would not purposefully recommend a maintenance plan that would not provide adequate protection.
As a side bar- It could be that manufacturers know most "conventional" oils today do in fact have some Group III and above base oil in the formulation. Due to this it is likely that the conventional oils perform much better than what they're tested for (in the eyes of the manufacturers), this, added with the cheaper cost-to-own conventional will provide that's been pushed my the rags and accountants could be the reason they do not recommend/require syn. At the end of the day, an auto manufacturer would not purposefully recommend a maintenance plan that would not provide adequate protection.