  We show that for any purely 2-unrectifiable metric space $M$, for example the
Heisenberg group $\mathbb{H}^1$ equipped with the Carnot-Carath\'{e}odory
metric, every homotopy class $[\gamma]$ of Lipschitz paths contains a length
minimizing representative $\gamma_\infty$ that is unique up to
reparametrization. The length minimizer $\gamma_\infty$ is the core of the
homotopy class $[\gamma]$ in the sense that the image of $\gamma_\infty$ is a
subset of the image of any path contained in $[\gamma]$. Furthermore, the
existence of length minimizers guarantees that only the trivial class in the
first Lipschitz homotopy group of $M$ with a base point can be represented by a
loop within each neighborhood of the base point. The results detailed here are
used in arXiv:2402.10420 to define and prove properties of a universal
Lipschitz path space over $\mathbb{H}^1$.
