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Gapless Weyl semimetals (WSMs) are a novel class of topological materials that host massless Weyl fermions as their low-energy excitations. When the Weyl cone is tilted, the Lorentz invariance is broken and the Lifshitz transition drives the system from type-I WSMs into type-II WSMs. Here, based on the lattice model, we perform a systematic numerical study of the effect of on-site disorder on the diagonal magnetoconductivity in tilted WSMs.We use the self-consistent Born approximation to calculate the disorder-induced self-energy and then apply the Kubo-Streda formula to calculate the diagonal magnetoconductivity. For the transverse magnetoconductivity σ_xx, the disorder is found to exhibit distinct effects in the quantum limit regime and in the quantum oscillation regime of type-I WSMs, which could be understood from Einstein's relationship.With strong disorder, σ_xx shows a linear relation with the inverse magnetic field 1/B , which exhibits certain robustness to both the Fermi energy and the Weyl cone tilting. For the longitudinal magnetoconductivity σ_zz , the strong disorder can break the positive magnetoconductivity as well as the Shubnikov-de Haas oscillations.By analyzing the spectral function, we find that the chiral anomaly is still preserved at strong disorder even when the Weyl cone is overtilted, as there is no gap opening around the Weyl nodes. The implications of our results for experiments are discussed.