Injectivity conjecture for harmonic models of semi-lopsided algebraic actions

Let $G$ be a countable discrete group with a left-invariant partial order $\preceq$. Let $f\in \mathbb{Z}(G)$ be semi-lopsided and satisfy

$$\operatorname{supp}(\widehat{f})\setminus\{1\}\subseteq\{g\in G:g\succ 1\}.$$

Set $m=\tau(f)$. Suppose either that $f$ has an $\ell^{2}$ formal inverse $\xi$ and $m$ is odd, or that $f$ has an $\ell^{1}$ formal inverse. In the respective cases, let $\Theta_{\xi}$ be the map defined in the corresponding case of the equal-entropy factor-map construction. Injectivity conjecture. The map $\Theta_{\xi}$ is injective modulo null sets. The preceding discussion shows that this factor map has the same entropy as the Bernoulli shift, making injectivity plausible; the source does not provide a resolution of the claim.