The graded homology conjecture for real loci of complex-conjugation-stable building sets

Let ${\cal G}$ be an $\mathbb R$-rational building set. A real subspace $C\in{\cal C}_{\cal G}$ is purely complex when it decomposes as

$$C=\bigoplus_i G_i\oplus \overline{G_i}.$$

Write $\mathbb R({\cal G})$ for the real subspaces in ${\cal G}$, and for a real $A\in{\cal C}_{\cal G}$ let $\mathbb C(A)$ be the sum of the complex components of $A$. The homology of the real locus should be naturally graded by $\mathbb R({\cal C}_{\cal G})$, with an isomorphism

$$2H_*({\overline Y}_{\cal G}(\mathbb R),\mathbb Z)[A]\cong\!\!\!\bigoplus_{\substack{\mathbb C(A)\subset C\subset A\C\text{ purely complex}}}\!\!\!H_*({\overline Y}_{{\cal G}|_C}(\mathbb R),\mathbb Z)[C]\otimes 2H_*({\overline Y}_{\mathbb R({\cal G}/C)}(\mathbb R),\mathbb Z)[A/C].$$

This conjecture extends the real-building-set homology description to building sets closed under complex conjugation. The complex case is more difficult because the cell structure does not carry over and operad maps may fail to preserve grading; no resolution is given here.