Constant-norm geodesics are shifted exponential solutions

Let $\mathcal{M}(n)$ denote the vector space of $n\times n$ matrices equipped with the Hilbert--Schmidt norm, and consider geodesics on $\mathsf{SL}(n)$ with the Hilbert--Schmidt metric. A shifted exponential solution is a curve of the form

$$Be^{tC}$$

for constant matrices $B$ and $C$ for which the curve lies in $\mathsf{SL}(n)$.

Constant-norm geodesic conjecture. Shifted exponential solutions of the form $Be^{tC}$ are the only geodesics with constant norm when viewed as curves in the vector space $\mathcal{M}(n)$ with the Hilbert--Schmidt norm.

The supplied text gives no resolution status or further evidence for this conjecture, so it remains open in the database.