There are, as well you know, two readings for that sentence. What does it look like in Thai? According to Google, like this:
สุนัขแมวทุก chases บาง
Thukkhn_EVERY rạk_LOVES s̄unạk̄h_DOG mæw_CAT bāng_SOME
That looks non-projective to me. But a similar sentence might never have appeared in the training data. So let’s try something a bit simpler.”John loves Mary” and “Mary loves John” both come out as Cxh̄̒n mærī̀ rạk. “John loves to fish” is Cxh̄̒n chxb plā and “John loves fish” is Cxh̄̒n rạk plā. Don’t tell Mary.