Some moduli stacks of symplectic bundles on a curve are rational

Let C be a smooth projective curve of genus g⩾2 over a field k. Given a line bundle L on C, let Sympl2n,L be the moduli stack of vector bundles E of rank 2n on C endowed with a nowhere degenerate symplectic form up to scalars. We prove that this stack is birational to BGm×As for some s if deg(E)=n⋅deg(L) is odd and C admits a rational point P∈C(k) as well as a line bundle ξ of degree 0 with ξ⊗2≇OC. It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.