The orbit-invariance conjecture for horospherical varieties

Let $X$ be a horospherical complexity-zero variety, let $\tau$ be a face of its cone $\sigma^\vee$, and let $O_\tau$ be the corresponding $G$-orbit. Let $p_1,\ldots,p_k$ be the primitive vectors on the rays $\rho_1,\ldots,\rho_k$ of $\sigma$ that are normal to $\tau$. For $1\leq i\leq k$, define

$$\xi_i=\sigma^\vee\cap\langle p_1,\ldots,p_{i-1},p_{i+1},\ldots,p_k\rangle^\bot.$$

Here a $\tau$-root with distinguished ray $\rho_a$ is a $\tau$-root whose distinguished ray is $\rho_a$. The orbit-invariance conjecture. The closure of the $G$-orbit $O_\tau$ is not $\operatorname{AAut}(X)$-invariant if and only if both of the following conditions hold: there exists $1\leq a\leq k$ and a $\tau$-root with distinguished ray $\rho_a$, and points of $O_\tau$ and $O_{\xi_a}$ have equal dimensions of tangent spaces $T_xX$.

This conjecture characterizes when a $G$-orbit closure fails to be invariant under the subgroup generated by additive group actions. It addresses the gap between the existence of a combinatorial $\tau$-root and the existence of a corresponding homogeneous locally nilpotent derivation; the supplied context does not state whether the conjecture is proved or remains open.