Geometric fixed points of norms via the normalizer

Let $G$ be a compact Lie group, let $K<G$ be finite, and let $N$ be the normalizer of $K$ in $G$. Let $X$ be an input for the norm $N_e^G$. Normalizer geometric fixed-point conjecture. There is a natural non-equivariant weak equivalence

$$(N_e^G X)^{\Phi K}\simeq \int_{N\backslash G}\int_{K\backslash N\times T_e(N\backslash G)}X.$$

This is the proposed replacement for factorization homology over $K\backslash G$ when that quotient is not parallelizable; the normalizer decomposition supplies the required framed pieces.