Stanley sequences with odd character

Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a well-structured class of Stanley sequences with two main parameters: the character $lambda(A)$ and the repeat factor $rho(A)$. Rolnick conjectured that for every $lambda in mathbb{N}_0backslash{1, 3, 5, 9, 11, 15}$, there exists an independent Stanley sequence $S(A)$ such that $lambda(A) =lambda$. This paper demonstrates that $lambda(A) otin {1, 3, 5, 9, 11, 15}$ for any independent Stanley sequence $S(A)$.