Reconstruction of a linear recurrence of maximal period over a Galois ring of characteristic $p^3$ by its highest digital sequence

Sequences $w$ over a field $GF(q)$, $q=p^r$, $p>2$, obtained by highest digit sequence of linear recurrent sequences $u$ over a Galois ring $R=GR(q^3,p^3)$ in some digit set are considered. The conditions guaranteeing the uniqueness of reconstruction of $u$ given $w$ is studied.