Kazarian's rational homotopy conjecture for the Kazarian space

Let $\tau$ be a collection of singularity types, let $X_\tau$ be the classifying space for $\tau$-cobordisms, let $\mathcal K_\tau$ be the Kazarian space, let $\mathcal K^+_\tau$ be its disjoint union with a point, and let $\Gamma(A)=\Omega^\infty S^\infty(A)$. For a virtual complex $A$, write $\cong_{\mathbb Q}$ for rational homotopy equivalence. Kazarian's conjecture.

$$X_\tau \cong_{\mathbb Q} \Gamma S^k(\mathcal K^+_\tau).$$

The conjecture relates the classifying space of $\tau$-cobordisms to the Kazarian space. The paper subsequently proves this rational homotopy equivalence, so the conjecture is solved.