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We focus on the slow-branching regime, where particles move at rate one and branch at rate $\\lambda\\in(0,1)$. For $b>2$, we show that $\\tau_{\\mathrm{cov}}(d)=x_\\star d+\\lambda^{-1}\\log d+O_{\\mathbb P}(1)$. For $b=2$, we show that $\\tau_{\\mathrm{cov}}(d)=x_\\star d+\\chi^{-1}\\log\\log d+O_{\\mathbb P}(1)$. Here, $x_\\star$ and $\\chi$ are explicit positive constants depending only on $b$ and $\\lambda$. 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