Stable-perturbation conjecture for bounded cone orbits

Let $d\geq 3$, let $X_d=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, and let $\mathfrak{a}$ be the diagonal subalgebra of $\mathfrak{sl}_d(\mathbb{R})$. For an open cone $C\subseteq\mathfrak{a}$, let $A_C$ be its image under the exponential map. A bounded $A_C$-orbit is an $A_C$-stable perturbation of a compact $A$-orbit if it has the form $x=gy$, where $g\in H^{0-}(A_C)$ and $Ay$ is compact, with

$$H^{0-}(A_C)=\bigcap_{a\in A}\{g\in G:\lim_{n\to+\infty}a^nga^{-n}\ \text{exists}\}.$$

Stable-perturbation conjecture. For every open cone $C\subseteq\mathfrak{a}$, every bounded $A_C$-orbit in $X_d$ contains an $A_C$-stable perturbation of a compact $A$-orbit. This is presented as a refined form of Margulis' conjecture for general cones, and no resolution is given.