The successive-overrelaxation (SOR) iterative method for linear systems is well understood if the associated Jacobi matrix B is consistently ordered and weakly cyclic of index 2. If, in addition, B 2 has only nonnegative eigenvalues and if ϱ( B), the spectral radius of B, is strictly less than unity, then by D. M. Young's classical theorem, the optimal relaxation parameter for the SOR method is given by ω b≔ 2 1+ 1−ϱ 2(B) . Young derived this result assuming that σ ( B 2) ⊂ [0, β 2] (with β = ϱ( B)) (∗) is the only information available about the spectrum σ( B 2) of B 2. It is also well known that no polynomial acceleration can improve the asymptotic rate of convergence of the SOR scheme if the optimal relaxation parameter has been selected. The recent claim by J. Dancis “that a smaller average spectral radius can be achieved by using a polynomial acceleration together with a suboptimal relaxation factor ( ω < ω b )” therefore comes as a surprise. A closer look however reveals that this improvement can only be achieved if more profound information on σ( B 2), of the form σ( B 2) ⊂ [0, γ 2] ∪ { β 2} (with γ < β), (∗∗) is at hand. We show that no polynomial acceleration of the SOR method (for any real ω) is asymptotically faster than the SOR scheme with ω = ω b under the assumption 08 (∗), thereby answering the question in the title of this paper in the affirmative, as well as solving an old related conjecture of D. M. Young. We also carefully investigate the question of what can be gained from the additional information (∗∗).