Uniqueness conjecture for s-deep Zero Linear Recurrence Sequences with a specified recurrence

Let $\{G_n\}_{n=1}^{\infty}$ be an $s$-deep Zero Linear Recurrence Sequence with recurrence relation

$$G_{n+1}=G_{n-2}+cG_{n-3},$$

where $c\geq 4$.

Uniqueness conjecture. There exists a unique decomposition for each positive integer $N$.

This conjecture addresses the existence of unique legal decompositions in a particular family of $s$-deep Zero Linear Recurrence Sequences. It is presented as an empirical conjecture in the supplied text, with no resolution stated there.