Type 2 growth for irregular Stanley sequences

Let $S(A)=\{a_n\}$ be a Stanley sequence, and call it irregular if it is not regular. Type 2 growth refers to the asymptotic pattern

$$a_n\approx c'\frac{n^2}{\log n}.$$

Type 2 growth conjecture. Every irregular Stanley sequence follows Type 2 growth. The paper proves that regular Stanley sequences follow Type 1 growth and identifies this assertion as the unresolved complementary part of the proposed regular/irregular dichotomy.