A while back, there was a question in class about DPs that seem to be able to have adverbs in them, like the subject of John’s skillfully baking cakes amused me.
I didn’t really provide a good answer to how to think about these in class, but I realized that this is actually covered in exercise 1 of the DP chapter in the textbook. It requires a little bit of “thinking outside the box” in a way. Without posting the answer (since it is a homework problem), there is a clue about what Adger had in mind in the statement of Task 4.