Let \(\ell\geq 5\) be a prime, \(\zeta\) a primitive \(\ell\)th root of unity in the algebraic closure of \(\mathbb Q_\ell\), \(K=\mathbb Q(\zeta)\), \(\lambda= 1-\zeta\), and \(\widehat{K}=\mathbb Q_\ell(\zeta)\) the \(\lambda\)-adic completion of \(K\). The Hilbert symbol \(( ,)_\lambda\) defines an orthogonality relation in \(\widehat{K}^\ast\). Let \(C\) denote the group of cyclotomic units of \(K\). A conjecture of \textit{G. Terjanian} [Acta Arith. 54, 87-125 (1989; Zbl 0642.12010)] asserts: If \(a\in\mathbb Z\setminus\ell\mathbb Z\) and \(a-\zeta\) is orthogonal to \(C\), then \(a\equiv\pm 1\bmod\ell\). The author investigates the following weaker conjecture: If \(a\in\mathbb Z\setminus\ell\mathbb Z\) and \(a^n-\zeta^n\) is orthogonal to \(C\) for all \(n\in\mathbb N\setminus\ell\mathbb N\), then \(a\equiv\pm 1\mod\ell\). This is shown to be true in many separate cases.