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We enable the estimation of the per-axon axial diffusivity from single encoding, strongly diffusion-weighted, pulsed gradient spin echo data. Additionally, we improve the estimation of the per-axon radial diffusivity compared to estimates based on spherical averaging. The use of strong diffusion weightings in magnetic resonance imaging (MRI) allows to approximate the signal in white matter as the sum of the contributions from axons. At the same time, spherical averaging leads to a major simplification of the modeling by removing the need to explicitly account for the unknown orientation distribution of axons. However, the spherically averaged signal acquired at strong diffusion weightings is not sensitive to the axial diffusivity, which cannot therefore be estimated. After revising existing theory, we introduce a new general method for the estimation of both axonal diffusivities at strong diffusion weightings based on zonal harmonics modeling. We additionally show how this could lead to estimates that are free from partial volume bias with, for instance, gray matter. We test the method on publicly available data from the MGH Adult Diffusion Human Connectome project dataset. We report reference values of axonal diffusivities based on 34 subjects, and derive estimates of axonal radii. We address the estimation problem also from the angle of the required data preprocessing, the presence of biases related to modeling assumptions, current limitations, and future possibilities.