Approximation of completed homology classes by sums of geodetic circles

Let $\widehat{H'_1}$ be the relevant completion of the homology group, and let $c\in\widehat{H'_1}$. A Cauchy sequence is a sequence converging in the associated metric completion. A geodetic circle is a circle for which, between every pair of points, one of its two arcs is a shortest path in the ambient metric space.

Geodetic approximation conjecture. For every $c\in\widehat{H'_1}$, there is a Cauchy sequence $(\chi_i)_{i\in\mathbb N}\in c$ such that every $\chi_i$ is the homology class of a finite sum of geodetic circles.

The statement proposes a stronger approximation property for completed homology classes. The supplied text does not establish it or give a resolution.