Recovery of finitely many biases for mean-zero non-symmetric random walks

Let $S$ be the random walk in Theorem 1, with finite-variance increments, and let $\theta$ be a configuration of finitely many biased coins whose bias function is denoted by $\eta$. Assume that the increments of $S$ have mean zero but are not symmetric.

Recovery conjecture. The bias function $\eta$ can be recovered up to a shift only.

This conjecture asks whether the recovery result for symmetric walks extends to mean-zero, non-symmetric walks, with reflection no longer among the unavoidable ambiguities. The paper states that the required algebraic generalization, in particular the corresponding full-rank lemma, had not been proved.