Special case of Margulis's conjecture

Let $D$ be the set of diagonal matrices in $\mathrm{SL}_k(\mathbb R)$, where $k \geqslant 3$. Let $z \in \mathrm{SL}_k(\mathbb R)/\mathrm{SL}_k(\mathbb Z)$, and suppose that $Dz$ has compact closure. Special case of Margulis's conjecture.

$$Dz \text{ is closed}.$$

The paper identifies this statement as equivalent to the preceding special case concerning products of linear forms, and attributes it to Margulis. It remains open in the stated generality.