Let $(V,Q)$ be a pair, with $V=\mathbb{F}_2^{2n}$ ($n\geq 2$) and $Q$ a nondegenerate quadratic form on $V.$ We consider the poset $\mathcal{P}_n$ of affine totally isotropic (with respect to $Q$) subspaces of $V$, ordered by inclusion. Does $\Delta(\mathcal{P}_n)$ have the homotopy type of a wedge of spheres? If so, and the dimension of the spheres is $r$, is it known $H_r(\Delta(\mathcal{P}_n),\mathbb{F}_2)$ as a representation of $V=(\mathbb{Z}/2)^{2n}$?

My guess comes from the simplest case: one can show by basic algebraic toplogy that $\Delta(P_2)\simeq\displaystyle\bigvee^{15} S^2$ ($Q=Q_+$, with $Q_+(x)=x_1x_2+\dots+x_{2n-1}x_{2n}$), but even in that case the action of $(\mathbb{Z}/2)^4$ on $H_2(\Delta(\mathcal{P}_2),\mathbb{F}_2)$ is mysterious to me. I am not sure if this poset and its properties are well known in the literature (like the whole poset of affine subspaces); I would really appreciate a suggestion.