``Hasse principle'' for \(\text{GL}_n(D)\) (Q1579691)

Let \(D\) be a Euclidean ring, \(n\in\mathbb{N}\) and \(\text{GL}_n(D)\) the \(n\)-th general linear group over \(D\). If \(f\) is an endomorphism of \(\text{GL}_n(D)\) which preserves conjugacy classes then \(f\) is already an inner automorphism.