Spectral uniqueness of bi-invariant metrics on symplectic groups

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{S}\mathrm{p}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{S}\mathrm{p}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is not right-invariant cannot be isospectral to a bi-invariant metric. The proof is elementary and uses a very strong spectral obstruction proved by Gordon, Schueth, and Sutton.