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Many applications of Nb/Al-AlOx/Nb Josephson junctions (JJs) in superconducting electronics require high-quality tunnel barriers with low subgap leakage that is usually characterized by figure of merit Vm = Ic Rsg, where Ic is the critical current and Rsg is the subgap resistance at 2 mV and 4.2 K. It is widely believed, and there is considerable literature suggesting, that quality and reproducibility of JJs critically depend on the intrinsic stress in Nb/Al-AlOx/Nb trilayers, and the stress therefore should be carefully minimized and controlled. Contrary to this belief, we show that JJ quality Vm and reproducibility do not depend on the stress in the trilayer, at least in the studied range from -300 to 300 MPa. In this range, Vm depends neither on the stress in a Nb/Al base electrode nor in a Nb counter electrode. We have found, however, that Vm crucially depends on the way the tunnel barrier formation by thermal oxidation of Al is done. For instance, room-temperature dynamic oxidation (in O2 flow at low pressures) in a cryopumped chamber leads to poor run-to-run reproducibility of Vm and reduced Vm values, whereas dynamic oxidation at the same parameters but in a chamber with a turbomolecular pump results in high Vm values and excellent run-to-run reproducibility.