The lexicographic characterization of the sequences $S_k$

Let $\mathcal{S}_k$ be the sequence written in base $3/2$, and let $S_k$ denote the result of interpreting that representation in base $3$. The lexicographic characterization conjecture. Sequence $\mathcal{S}_k$ written in base $3/2$ but interpreted in base $3$ is sequence $S_k$. The preceding discussion establishes that these interpreted sequences are 3-free; the conjectural point is their identification with the sequence $S_k$.