Continuous degree-two inflation: classes split by L are inflated

Let $K \subseteq \Omega$ be fields with $\Omega/K$ Galois, and let $r \colon (\Omega \simeq_{\mathrm{alg}[K]} \Omega) \to \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be a group homomorphism (a "level map"), subject to two cofinality hypotheses: hlevel, that for every intermediate field $E$ of $\Omega/K$ finite over $K$ there is an intermediate field $F$ of $\overline{\mathbb{Q}}/\mathbb{Q}$ finite over $\mathbb{Q}$ with $r^{-1}(\mathrm{Gal}(\overline{\mathbb{Q}}/F)) \subseteq \mathrm{Gal}(\Omega/E)$, and hopen, the converse, that for every such $F$ there is such an $E$ with $r(\mathrm{Gal}(\Omega/E)) \subseteq \mathrm{Gal}(\overline{\mathbb{Q}}/F)$. Let $L$ be an intermediate field of $\Omega/K$, finite-dimensional and normal over $K$, and let $c$ lie in the subgroup levelCocycles₂ r (Rep.ofAlgebraAutOnUnits K Ω) of $2$-cochains on $\mathrm{Gal}(\Omega/K)$ valued in the $\mathrm{Gal}(\Omega/K)$-module $\Omega^\times$ (written additively). Assume hres: the restriction of the underlying cochain of $c$ along the homomorphism $\mathrm{Gal}(\Omega/L) \cong \mathrm{Gal}(\Omega/L)^{\mathrm{fix}} \hookrightarrow \mathrm{Gal}(\Omega/K)$, obtained from IntermediateField.fixingSubgroupEquiv L followed by the inclusion of the fixing subgroup, belongs to levelCoboundaries₂ for the composed level map and the module $\Omega^\times$ over $L$. Then the image of $c$ in the quotient continuousH2 r (Rep.ofAlgebraAutOnUnits K Ω) of levelCocycles₂ by the preimage of levelCoboundaries₂, under the projection continuousH2π, lies in the subset of classes of the form continuousH2π r (Rep.ofAlgebraAutOnUnits K Ω) ⟨unitsInflate₂ L f, h⟩ for some $f \colon \mathrm{Gal}(L/K) \times \mathrm{Gal}(L/K) \to \mathrm{Additive}\, L^\times$ in cocycles₂ (Rep.ofAlgebraAutOnUnits K L) whose inflation unitsInflate₂ L f is itself a level cocycle.

This is the surjectivity half of inflation–restriction in degree two for the continuous (level-constant) cohomology of $\Omega^\times$: a class killed by restriction to $\mathrm{Gal}(\Omega/L)$ comes from a $2$-cocycle of $\mathrm{Gal}(L/K)$ with values in $L^\times$, as in the identification of $\mathrm{Br}(L/K)$ with $\ker(\mathrm{Br}(K) \to \mathrm{Br}(L))$. It is used in groupCohomology.exists_mem_split_adjoin_rootsOfUnity_of_padic, where classes over a $p$-adic field are exhibited as split by an explicit cyclotomic extension.