Geometric fixed-points of MO

Question

Recall that the value of the orthogonal spectrum $\mathbf{MO}$ at an inner product space $V$ is the Thom space of the tautological bundle over the Grassmannian of $|V|$-demensiomal planes in $V\oplus V$: $$\mathbf{MO}(V) = \text{Th}\left(\text{Gr}_{|V|}(V\oplus V)\right)=\{(x, U) \subset (V\oplus V) \times \text{Gr}_{|V|}(V\oplus V) : x \in U\} \cup \{\infty\}.$$ The structure maps are given by $$\sigma_{V,W}: S^V \wedge \mathbf{MO}(W) \longrightarrow \mathbf{MO}(V\oplus W), \quad \quad v \wedge (x, U) \longmapsto \bigg(\kappa_{V, W}\big(\underbrace{(v,0)}_{\in V\oplus V}, \ \underbrace{x}_{\in W\oplus W}\big), \kappa_{V, W}\big(\underbrace{(V\oplus 0)}_{\subseteq V\oplus V}\oplus\underbrace{U}_{\subseteq W\oplus W}\big) \bigg)$$ where $$\kappa_{V, W}: V\oplus V \oplus W \oplus W \longrightarrow V\oplus W \oplus V \oplus W, \quad \quad \quad (v,v',w,w')\longmapsto (v,w,v',w').$$ Let $G$ be a compact Lie group, $\mathcal{U}_G$ a fixed complete $G$-universe and $s(\mathcal{U}_G)$ the poset of finite dimensional subrepresentations of $\mathcal{U}_G$. I'm trying to calculate the geometric fixed-point homotopy groups of $\mathbf{MO}$, following Example 6.1.46 in Stefan Schwede's Global Homotopy Theory. By definition, $$\Phi^G_0 (\mathbf{MO}) = \underset{s(\mathcal{U}_G)}{\text{colim}}\ \left[ S^{V^G}, \ \mathbf{MO}(V)^G\right]_*$$ For a $G$-representation $W$ we write $W^\perp$ for $W - W^G$, where the minus sign means orthogonal complement. From the canonical decomposition $$ \text{Gr}_{n}(W)^G \cong \coprod_{k\geq 0} \text{Gr}_{k}(W^G) \times \text{Gr}_{n-k}(W^\perp)^G$$ we obtain $$ \text{Th}\left(\text{Gr}_{n}(W)\right)^G \cong \bigvee_{k \geq 0} \text{Th}\left(\text{Gr}_{k}(W^G)\right) \wedge \text{Gr}_{n-k}(W^\perp)^G_+.$$ Letting $W = V\oplus V$ we have $$ \bigvee_{j \in \mathbb{Z}}\mathbf{MOP}^{[j]}(V^G)\wedge \text{Gr}_{|V^\perp|+j}(V^\perp\oplus V^\perp)_+^G \cong \mathbf{MO}(V)^G,$$ where $\mathbf{MOP}^{[j]}$ is the orthogonal spectrum given by $$\mathbf{MOP}^{[j]}(U) = \text{Th}\left(\text{Gr}_{|U|+j}(U\oplus U)\right).$$ Finally, $$\Phi^G_0(\mathbf{MO}) \cong \bigoplus_{j\in \mathbb{Z}} \underset{s(\mathcal{U}_G)}{\text{colim}}\ \left[ S^{V^G}, \quad \mathbf{MOP}^{[j]}(V^G)\wedge \text{Gr}_{|V^\perp|+j}(V^\perp\oplus V^\perp)_+^G\right]_*, \tag{1}\label{1}$$ where the stabilization map for $V \subseteq W \subseteq \mathcal{U}_G$ is given by sending a class represented by $$f: S^{V^G} \longrightarrow \mathbf{MOP}^{[j]}(V^G)\wedge \text{Gr}_{|V^\perp|+j}(V^\perp\oplus V^\perp)_+^G$$ to the composite $$S^{W^G} \cong S^{(W-V)^G} \wedge S^{V^G} \xrightarrow{\text{id} \wedge f} S^{(W-V)^G}\wedge \mathbf{MOP}^{[j]}(V^G)\wedge \text{Gr}_{|V^\perp|+j}(V^\perp\oplus V^\perp)_+^G \\ \xrightarrow{\sigma_{(W-V)^G, V^G} \wedge \text{incl}_{V, W}^j} \mathbf{MOP}^{[j]}(W^G)\wedge \text{Gr}_{|W^\perp|+j}(W^\perp\oplus W^\perp)_+^G$$ where $$\text{incl}_{V, W}^j: \text{Gr}_{|V^\perp|+j}(V^\perp\oplus V^\perp)_+^G \longrightarrow \text{Gr}_{|W^\perp|+j}(W^\perp\oplus W^\perp)_+^G, \quad \quad L \longmapsto L + 0 \oplus\left( W^\perp - V^\perp\right).$$ My question is: What is the value of colimit \eqref{1} ? I'm hoping that it is the $\mathbf{MOP}^{[j]}$ homology of some Grassmannian, for example as in the case of $\mathbf{mO}$ in which we have $$ \Phi^G_*(\mathbf{mO}) \cong \bigoplus_{j\geq 0}\mathbf{mO}_{*-j}\left(\text{Gr}_j(\mathcal{U}_G^\perp)^G\right).$$