  It is a longstanding conjecture that given a subset $E$ of a metric space, if
$E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and
$\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is
an $\alpha$-rectifiable set. We prove this conjecture under the assumption that
the ambient metric space is a homogeneous group with a smooth-box norm.
