Finitely generatedness of continuously homogeneous indecomposable planar continua

A continuum is a compact connected metrizable space. A continuum is indecomposable if it cannot be written as the union of two proper subcontinua. A continuum is planar if it embeds in the plane. It is finitely generated if either it does not embed essentially into $\mathbb{S}^1\times I^\infty$, or it embeds essentially into $\mathbb{S}^1\times I^\infty$ and there exists a continuum $X^*$ in $\Sigma$ such that $\tau(X^*)=X$.

Finitely generatedness conjecture. Any continuously homogeneous indecomposable planar continuum is finitely generated.

This is the specific conjecture associated with the finitely generated class. The supplied text gives no resolution; it also notes examples of continua that are not finitely generated, including certain inverse limits and self-entwined circle-like continua.