Moduli of Conics on General Plucker Linear Sections of Grassmannians

Let $G=\operatorname{Gr}(k,V)$ be a complex Grassmannian in its Pl$ü$cker embedding, and let $Y_E$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics using the kernel and span of a conic. These define a flag variety $B=\operatorname{Fl}(k-2,k+2;