Identification conjecture for ranked groups with an almost self-normalizing TI-subgroup

Let $G$) be a connected ranked group with involutions but with no infinite elementary abelian subgroup. Suppose that $G$ has a definable, connected subgroup $C<G$ which is a TI-subgroup, meaning that its distinct conjugates intersect trivially, and is almost self-normalising. Suppose that $C$ has even index in $N=N_G(C)$. $A_1$-Conjecture. Then

$$G\simeq \operatorname{PGL}_2(\mathbb{K}).$$

This conjecture proposes that the involution geometry associated with an almost self-normalizing TI-subgroup characterises the characteristic-not-$2$ form of the root system $A_1$, potentially extending beyond the finite Morley rank setting; its resolution is not supplied here.