The relative Grothendieck conjecture for pro-class Galois groups

Let $K_i$ be number fields for $i=1,2$, let ${\mathcal C}_i$ be nontrivial full classes of finite groups, and let

$$\sigma:G_{K_1}^{{\mathcal C}_1}\overset{\sim}\to G_{K_2}^{{\mathcal C}_2}$$

be an isomorphism. Here $\Sigma({\mathcal C}_i)$ denotes the associated set of primes, and $K_i$ has a complex prime when it has a complex archimedean place. Assume that, for one $i$, either $2\in\Sigma({\mathcal C}_i)$ or $K_i$ has a complex prime. The relative Grothendieck conjecture. There exists a unique

$$\tau\in\operatorname{Iso}(K_2^{{\mathcal C}_2}/K_2,K_1^{{\mathcal C}_1}/K_1)$$

such that $\sigma$ coincides with the isomorphism induced by $\tau$. This is an anabelian reconstruction claim for number fields from their pro-${\mathcal C}$ Galois groups; the supplied text does not state evidence that it has been resolved, so its status is left open.