I read nothing of problems presented which require both types of thinking. Common problems involving both like seating arrangements for a wedding or which family member to ask to borrow money could be used for this. In the test discussed in the article the participant is going to catch on to the pattern after a few questions and instinctively switch their thinking to an optimal mode. In my opinion the resulting brain activity they're reading isn't empathy/analytics, it's bound and unbound thought. Empathy isn't imagination. It's the experiences, memories, and emotions of oneself and everyone they've known recalled abstractly. Asking someone to answer "social question" without personal context not only unbounds the process, but simultaneously removes analysis. This is caused by a) the need for one to imagine the contextual characters necessary to fill in the gaps and b) the inherent throw-away nature of those virtual characters. It all sits in short-term memory.
Real empathy requires actual people to empathize with because it involves more than words. It includes body language and so many other factors. All that's been invoked here is structured imagination, and these questions would inherently exclude analytic thought in that test characters must be taken as-is. On the other side, analytic thought requires mental sandboxes. Analyzing a hypothetical question presents it's own sandbox which excludes imagination, and relies solely on ones training & short term memory. The fact that the subjects know they are being tested at all seems to be the originating flaw.