  We show that every $\mathbb{R}^d$-valued Sobolev path with regularity
$\alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided
$\alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying
Hairer's reconstruction theorem generalized to a framework allowing for Sobolev
models and Sobolev modelled distributions. Moreover, we show that the
corresponding lifting map is locally Lipschitz continuous with respect to the
inhomogeneous Sobolev metric.
