Character range conjecture for Stanley sequences

For a regular Stanley sequence $S(A)$, let $\lambda(A)$ denote its character. The range of the character function is the set of values attained by $\lambda(A)$ over regular Stanley sequences. Character range conjecture. This range is exactly

$$\{n\in\mathbb{Z}: n\ge 0\}\setminus\{1,3,5,9,11,15\}.$$

The claim is based on computational evidence: the six excluded values were found impossible in the examined cases, while all other values up to $76$ were realized; the source notes that the asserted impossibility was not definitive because irregularity had not been ruled out for certain sequences.