Specialness conjecture for affine spaces associated with inflated conjugacy classes

Let $G$ be an algebraic group, let $P$ be a parabolic subgroup with unipotent radical $N$, and let $C$ be a conjugacy class in $P/N$. Let $D$ be the inflation of $C$ to $P$, and let $N^C$ be the largest normal subgroup of $P$ contained in $N$ such that, for every $\gamma\in D$, all elements of $D\cap\gamma N^C$ have the same canonical parabolic. For $\delta\in C$, write $P_{\delta N}$ for the connected stabiliser of $\delta N$ in $P$. Specialness conjecture. For any $\gamma\in D$, the affine space $\gamma N/N^C$ is $P_{\delta N}$-special. The conjecture relates inflated conjugacy classes, canonical parabolic subgroups and special homogeneous spaces; the supplied text does not state whether it has been proved or remains open.