Finite-filling shapes conjecture for the free group

Let $F_2$ be the free group on generators $a$ and $b$, and let $S\subset F_2$ be a shape. Say that $S$ has finite filling when the solitaire filling process generated by $S$ can fill only finitely many sites. A cyclic subgroup is a subgroup of the form $\langle g\rangle$, and a coset of it is a set of the form $x\langle g\rangle$. Finite-filling shapes conjecture. The shape $S$ has finite filling if and only if it is not contained in a coset of a cyclic subgroup $\langle g\rangle$.

The preceding theorem proves the corresponding criterion for connected shapes, while the conjecture asserts the stated characterization for arbitrary shapes in $F_2$. The authors explain that this is expected by analogy with $\mathbb{Z}^d$, where the finitely filling shapes are the non-linear ones, but they do not have a proof.