Rolnick's character-range conjecture for independent Stanley sequences

Let $S(A)=(a_n)$ be an independent Stanley sequence: there are constants $\lambda=\lambda(A)$ and $\kappa=\kappa(A)$ such that for all $k\ge\kappa$ and $0\le i<2^k$, $a_{2^k+i}=a_{2^k}+a_i$ and $a_{2^k}=2a_{2^k-1}-\lambda+1$. The constant $\lambda$ is called the character of $S(A)$. Rolnick's character-range conjecture. The range of the character function is exactly the set of nonnegative integers not in $\{1,3,5,9,11,15\}$. The conjecture concerns which characters occur among independent Stanley sequences; the source reports constructions for every character up to $75$ except the six listed values and gives no resolution.