Existence of Theories Which Satisfy The Axioms
One can generalize the Wightman axioms to dimensions other than 4. In dimension 2 and 3, interacting (i.e. non-free) theories which satisfy the axioms have been constructed.
Currently, there is no proof that the Wightman axioms can be satisfied for interacting theories in dimension 4. In particular, the Standard model of particle physics has no mathematically rigorous foundations. There is a million dollar prize for a proof that the Wightman axioms can be satisfied for gauge theories, with the additional requirement of a mass gap.
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