There are, as well you know, two readings for that sentence. What does it look like in Thai? According to Google, like this:
??????????? chases ???
Thukkhn_EVERY r?k_LOVES s?un?k?h_DOG m?w_CAT b?ng_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 Cxh??n m?r?? r?k. “John loves to fish” is Cxh??n chxb pl? and “John loves fish” is Cxh??n r?k pl?. Don’t tell Mary.