McL conjecture on algebraic characterisation of planar Peano continua

Let $X$ be a compact, locally connected, metrizable space that is locally embeddable in $S^2$. A simple set of circles in $X$ is a set $S$ of circles with the simplicity property used in the definition of the corresponding homology group. For a metric $d$ inducing the topology of $X$, let $\hat{H}_d(X,d)$ denote the associated homology group, and for a circle $\chi$ let $\llbracket \chi \rrbracket$ be its corresponding element. Then McL conjecture. $X$ 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

$$U:=\{\llbracket \chi \rrbracket\in\hat{H}_d(X,d)\mid\chi\in S\}$$

spans $\hat{H}_d(X,d)$. This conjecture gives an algebraic characterisation of the Peano continua embeddable in the plane, analogous to the cited characterisation theorem. The source notes strong evidence, including the Sierpiński triangle, for which the triangular face boundaries can be chosen as $S$; its resolution is not stated.