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Generation of electron-positron pairs via the multiphoton Breit-Wheeler process in an all-optical scheme will be made possible on forthcoming high-power laser facilities through the collision of wakefield-accelerated GeV electrons with a counter-propagating laser pulse of 10^{22}–10^{23} W cm^{-2} peak intensity. By means of integrated 3D particle-in-cell simulations, we show that the production of high-density sources of ultrarelativistic electron-positron pairs is within the reach of soon-to-be-available laser systems. Under physical conditions accessible to the dual-beam CILEX-Apollon facility, we find that the generated positrons can carry a total charge of 0.05–1 nC, with a mean energy of 100–400 MeV and an angular divergence of 0.01–0.1 rad. The variations of the positron source’s properties with respect to the laser parameters are also examined.