\left(\mathcal{A}^0_F(r(\Lambda_g)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right)$ and $\left(\tilde{\mathcal{A}}^0_F(r(\Lambda_g)), (\tilde{X}^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right)$ 