Sublogarithmic growth conjecture for the bounded-exponent semigroup

Let $A(x)=\#\bigl(\mathcal{S}_3^{(\leq 3)}\cap[1,x]\bigr)$, where $\mathcal{S}_3^{(\leq 3)}$ is the bounded-exponent semigroup associated with the prime set $\mathcal{P}_3$. Sublogarithmic growth conjecture.

$$A(x)=o(\log x)\qquad\text{as }x\to\infty.$$

The paper describes this as a very strong sufficient condition for the desired finiteness result, and notes that it conflicts with the empirical growth suggested for $\mathcal{P}_3$. It remains an open, highly implausible auxiliary conjecture rather than the paper’s preferred route.