Near-modular set conjecture for realizing all even independent Stanley characters

A set $A$ is near-modular modulo $3^{\ell+1}$ if it has the near-modularity property used in the paper. Near-modular set conjecture. For every $\ell\in\mathbb{Z}^{+}$, there exists a near-modular set $A$ modulo $3^{\ell+1}$ such that

$$|A|=2^{\ell+1},$$

and the largest element of $A$ is either $2\cdot 3^{\ell}$ or $4\cdot 3^{\ell}$.

The paper states that proving this would suffice to show that all even characters can be achieved by independent Stanley sequences. The conjecture is presented as an unresolved final step.