Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: $\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$ as the wedge sum $\bigvee_{i \leq n} (\Sigma^\infty_+\mathrm{Sur}(\mathbb{R}^i,\mathbb{R}^{m_1}) \wedge \Sigma^\infty_+\mathrm{Sur}(\mathbb{R}^{n-i},\mathbb{R}^{m_2}) \wedge \Sigma^\infty_+ O(n)^\vee)_{h(N(O(i) \times O(n-i)))} \wedge \mathrm{Ad}^{O(n)}.$
Here $ N(O(i) \times O(n-i))$ is the normalizer in $O(n)$ and $\mathrm{Ad}^{O(n)}$ is the adjoint representation of $O(n)$. My question is whether or not this splitting actually exists.
It seems possible that this splitting is equivalent to Miller's stable splitting of Stiefel manifolds, and there are similar formulas one can write down for any partition $m_1+m_2+\dots +m_k$ which are also predicted to be true. Note that a similar formula does hold if we replace linear surjections by surjections of finite sets.
