Isnt it usual for a quantum field to have a conceptual problem?
Yes, quantum fields are quantum fields, but the rigor of mathematics and any failure should be explicitly clarified. What lies in this question is that quantum fields do not need to obey mathematical representation rigorously. In my paper I take a point of view of investigating with the rigor of mathematical reasoning. It could be reasonable to have flexible physical interpretations, but it is not a good reason to claim its rigorousness or dismiss a rigorous argument.