Reflection-positive invertible field theory classification conjecture

Let $H_n$ be the symmetry group in dimension $n$, let $MTH$ be the Thom spectrum associated to this symmetry, and let $I\mathbb Z$ denote the Anderson dual of the sphere spectrum. Reflection-positive classification conjecture. There is a one-to-one correspondence

$$\left\{\text{deformation classes of reflection-positive invertible $n$-dimensional extended field theories with symmetry group $H_n$}\right\}\cong [MTH,\Sigma^{n+1}I\mathbb Z].$$

This conjecture incorporates non-topological invertible theories into the classification by generalized cohomology; the source does not state a resolution.