A computational study of the inception of positive wire-cylinder corona discharges in low-pressure air in crossed electric and magnetic fields is performed. The inception voltages are calculated for a wide range of gas densities, wire radii, and applied magnetic fields. Conditions are considered when the reduced electric fields at wire electrodes reach extremely high values of about 10 kTd. An expression applicable at such strong fields for the ionization coefficient, which is a key parameter of the corona inception model, is presented against the values of electric and magnetic fields. Calculated inception voltages agree with a large quantity of available experimental data on low-pressure positive corona discharges, obtained both with and without the application of magnetic fields. The calculation results describe specific details of the non-monotonous dependence of the inception voltages on the magnetic field values, similar to those obtained in experiments.