Nonexistence of bounded geodesics on odd-dimensional special linear groups

Let $n$ be an odd positive integer, and equip the Lie group $\mathsf{SL}(n)$ with the Hilbert--Schmidt metric. A geodesic is bounded if its image is bounded in the ambient matrix space with the Hilbert--Schmidt norm.

Nonexistence conjecture. If $n$ is odd, then $\mathsf{SL}(n)$ has no bounded geodesics.

This remains open even for the block-diagonal solutions considered in the paper. The block-diagonal case is settled for $n=3$, but for $n\geq 5$ only sufficient conditions for non-compact geodesics are known.