Kitaev's conjecture on the spectrum of invertible phases

For each spatial dimension $d$, let $\mathcal I(\mathbb Z^d)$ be the abelian group of invertible gapped phases, and let $\mathrm{IP}(\mathbb Z^d)$ denote a proposed space-level construction of invertible states. Kitaev's conjecture. There is a canonical isomorphism

$$\pi_0(\mathrm{IP}(\mathbb Z^d))\cong \mathcal I(\mathbb Z^d).$$

Moreover, the spaces assemble naturally into an $\Omega$-spectrum $\mathbb{IP}=\{\mathrm{IP}(\mathbb Z^d)\}_{d\ge 0}$, equivalently with homotopy equivalences

$$\mathrm{IP}(\mathbb Z^d)\simeq \Omega(\mathrm{IP}(\mathbb Z^{d+1})).$$

The conjecture proposes a stable-homotopy-theoretic organization of invertible phases. Substantial evidence and partial constructions are known, but the full spectrum-level statement remains open.