MacLane-type planarity criterion for compact one-dimensional continua

Let $X$ be a compact, one-dimensional, locally connected, metrizable space with no cut point. A set $S$ of circles in $X$ is simple if every $1$-simplex of $X$ has a sub-simplex homotopic to it in at most two elements of $S$. Let $H_{\ell}(X)$ be the metric homology group associated with a metric $d$ inducing the topology of $X$, and let $U=\{\overline{\chi}\in H_{\ell}(X):\chi\in S\}$. The set $U$ spans $H_{\ell}(X)$ when the closure of its finite sums is all of $H_{\ell}(X)$.

MacLane-type planarity conjecture. The space $X$ is embeddable in $S^2$ if and only if there are a simple set $S$ of circles in $X$ and a metric $d$ inducing the topology of $X$ such that $U$ spans $H_{\ell}(X)$.

This is intended as an algebraic characterization of planar continua analogous to MacLane's criterion for finite graphs. Local connectedness and the no-cut-point hypothesis are necessary, as illustrated by counterexamples in the source; the conjecture itself remains open.