The $SL(n,\mathbb R)/SL(m,\mathbb R)$ cocompact-discontinuous-group conjecture

For integers $n>m$, consider the homogeneous space

$$SL(n,\mathbb R)/SL(m,\mathbb R).$$

Kobayashi's special-case conjecture. This homogeneous space does not admit a cocompact discontinuous group. This is presented as a special case of the compact-standard-quotient conjecture. The source supplies no separate resolution status for this special case.