In the area of superconducting spintronics, spin-triplet supercurrents in half-metallic ferromagnets (HMFs) could yield dissipationless spin transport over large distances and high current density. Promising among the HMFs is the perovskite oxide La0.7Sr0.3MnO3 (LSMO), and recent studies in combination with the high-Tc superconductor YBa2Cu3O7, or the conventional superconductor NbTi, showed long-range effects. Here we focus on two issues that as yet have received less attention: the value of the critical current in the HMF in the limit of a very small electrode distance (20 nm) and the nature of the spin-triplet generator. We use lateral junctions shaped as a bar, square, and disk, and find high supercurrent densities of the order of 1011 A/m2, pointing to an efficient triplet generation mechanism. This is surprising in the sense that no magnetic inhomogeneity is purposely built in, as is done in conventional metal triplet junctions. Furthermore, from the magnetic field dependence of the critical current interference patterns, we find a uniform supercurrent distribution in bar-shaped devices, but one more constricted to the rim in disk devices, which is an expected consequence of the geometry. We also analyze the temperature dependence of the critical current and find the quadratic dependence that was predicted in the limit of small junction lengths. From studying the NbTi/LSMO interface with scanning electron transmission microscopy, we conclude that the magnetic inhomogeneity required for triplet generation resides in the LSMO layer adjacent to the interface. Published by the American Physical Society 2024