Normal dilatation of triangular matrices

Let $R$ be a (real or complex) triangular matrix of order $n$, say, an upper triangular matrix. Is it true that there exists a normal $n\times n$ matrix $A$ whose upper triangle coincides with the upper triangle of $R$? The answer to this question is “yes” and is obvious in the following cases: (1) $R$ is real; (2) $R$ is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries of $R$ belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrix $R$ of order 2. However, even for $n=3$ this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for $3\times3$ matrices.