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A system model has been developed for the analytical and numerical treatment of the Wheeler cap technique. The model projects the properties of the antenna-under-test on the properties of the cavity, and was used in part I of the study to examine the performance of constant-size rectangular cavities. E/M simulations verified the model. Part II proceeds in the same framework and completes this two-part study, which estimates the limits of resonance-free efficiency measurements of planar antennas inside fixed-geometry Wheeler caps through a unified deterministic design methodology for 3 types of cavities. The focus is on the analytical and numerical treatment of spherical and cylindrical cavities. Spherical caps can provide coherent measurable bandwidths (BWs) in the range 28-41% for antenna sizes 0.5-0.85 times the wavelength. Cylindrical cavities are also versatile performers and display coherent measurable BWs in the range 20-40% for antenna sizes 0.5-0.78 times the wavelength. Electrically small antennas can secure measurable BWs up to 87% in the spherical case and up to 76% in the cylindrical one, when the cavities are operated below cut-off.