Essential one-to-one conjecture for generalized Bernoulli algebraic actions

Let $\Gamma$ be a countable group and let $[\,\cdot\,]\colon\Gamma\to\mathbb{Z}$ be a homomorphism. Suppose that

$$f=M-\sum_{s\in I} f_s s,$$

where $M>\sum_{s\in I}f_s$, and $[s]\geqslant1$ and $f_s>0$ for every $s\in I$. Let $Y$ be the Bernoulli source space, $X_f$ the associated algebraic action, and $\phi\colon Y\to X_f$ the homoclinic map. Essential one-to-one conjecture. The map $\phi$ is a measurable isomorphism. The preceding discussion establishes that $\phi$ is surjective and maps the Bernoulli measure to Haar measure; essential one-to-one-ness is not obvious and is presented as likely, so the assertion remains open.