The cosmall-root characterization conjecture for simply laced flag varieties

Let $G$ be a semisimple algebraic group with root system $R$, let $P$ be a parabolic subgroup with positive roots $R^{+}_P$, and let $1.P$ denote the base point of the flag variety $X=G/P$. For a degree $d\in H_2(X;\mathbb{Z})$ and a Schubert variety $Y\subset X$, write $\Gamma_d(Y)$ for its degree-$d$ curve neighborhood. A root $\alpha\in R^+\setminus R^{+}_P$ is $P$-cosmall if it is a maximal root of the degree $\alpha^\vee+\mathbb{Z}\Delta_P^\vee$.

Cosmall-root characterization conjecture. Assume that $R$ is simply laced and let $\alpha\in R^+\setminus R^{+}_P$. Then $\alpha$ is $P$-cosmall if and only if

$$\Gamma_{\alpha^\vee}(1.P)=X(s_\alpha).$$

This conjecture characterizes $P$-cosmall roots through the curve neighborhoods of the base point in a homogeneous space. Its status is not resolved in the supplied source context.