Growth conjecture for primary Carmichael numbers

Let $\mathcal{C}'$ be the set of squarefree numbers $m$ such that $s_p(m)=p$ for every prime factor $p$ of $m$, and let $C'(x)$ denote the number of elements of $\mathcal{C}'$ less than $x$.

Growth conjecture for primary Carmichael numbers. For sufficiently small $\varepsilon>0$,

$$C'(x)=O\left(x^{1/3-\varepsilon}\right)\quad\text{as }x\to\infty.$$

The conjecture is motivated by numerical evidence and by the expected rarity of Carmichael numbers with more than three prime factors. The related conjectural estimate of Granville and Pomerance for Carmichael numbers with exactly three prime factors is $O(x^{1/3}/\log^3 x)$, while the best bound mentioned in the source is weaker.