We present evidence that numerically accurate quantum calculations employing modern internucleon forces do not reproduce the proton analyzing power, A(y), for p- 3He elastic scattering at low energies. These calculations underpredict new measured analyzing powers by approximately 30% at E(c.m.) = 1.20 MeV and by 40% at E(c.m.) = 1.69 MeV, an effect analogous to a well-known problem in p-d and n-d scattering. The calculations are performed using the complex Kohn variational principle and the (correlated) hyperspherical harmonics technique with full treatment of the Coulomb force. The inclusion of the three-nucleon interaction does not improve the agreement with the experimental data.