For a given set \(X=\{x_ k\}_ 0^{n+1}\) of nodes in [-1,1], the author introduces an operator \(A_ n(X;x)\) mapping functions in C[-1,1] into polynomials of degree n. They were called `generalized alternating polynomials'. Here X is specialized to \(U=\{\cos k\pi /n+1\}_ 0^{n+1}\), the extrema of Chebyshev polynomials of the first kind. Writing \(A_ n(U;x)=\sum^{n+1}_{0}f(x_ k)a_ k(U;X)\) in the ``Lagrangian'' form, he is interested in the norm of the operator \(\| A_ n\|\). This operator was considered by \textit{E. W. Cheney} and \textit{T. J. Rivlin} [Math. Z. 145, 1, 33-42 (1975; Zbl 0295.41003)]. They compared it with the norm of the Lagrange interpolant and showed that \(\| A_{2n}(U)\| \geq \| L_{2n+1}(U)\|,\) where \(L_{2n+1}(U;x)\) denotes the Lagrange interpolant of degree \(2n+1\) on the U-nodes. They conjured that perhaps ``this is so also true for the odd case''. Here the author settles the conjecture in the affirmative and proves that \(\| A_ n(U)\| \geq \| L_{n+1}(U)\|\) for all n.