The motivic class of the space of genus $0$ maps to the flag variety

Let $\mathbb{k}$ be a field, let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}_{\mathbb{k}}$, and let $Ω^{2}_β(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=β$. We show that und