My question has been answered in Theorem 3.1 in https://arxiv.org/abs/2404.09023 

Indeed, $\pi_{D}\left(\left(\Omega^{d+1}X\right)^{\mathbb{Z}_2},c_{x_0}\right)\cong\pi_{D+1}\left(\Omega^{d}X,\left(\Omega^{d}X\right)^{\mathbb{Z}_2},c_{x_0}\right)$ for any $\mathbb{Z}_2$-space $X$ with $\mathbb{Z}_2$-fixed point $x_0\in X^{\mathbb{Z}_2}$ which gives rise to the long exact sequence of the pair $\left(\Omega^{d}X,\left(\Omega^{d}X\right)^{\mathbb{Z}_2}\right)$. The map $c_{x_0}$ denotes the constant map to $x_0$.