The conjecture that the Stanley sequence from {0, 4} has subquadratic lower growth

Let $A_4$ be the Stanley sequence starting from the set $\{0,4\}$. Its growth rate is compared with the scale $n^2/\log n$. The conjecture for $A_4$. The sequence does not have growth rate $\Theta(n^2/\log n)$; instead, it satisfies the upper bound $O(n^2/\log n)$, while its lower bound is approximately $\Omega(n^{2-\delta})$ for some $\delta>0$, meaning that the infimum of the values of $\delta$ for which this lower bound holds is strictly positive. This challenges the heuristic prediction for irregular Stanley sequences based on random behavior. The broader question of the growth rate of Stanley sequences starting from $\{0,n\}$ remains open, and the stated claim is supported by numerical evidence rather than a proof.