The growth conjecture for prime and thickened prime components

Let $C_{{\mathcal{P}^{\operatorname{pr}}}}(X)$ be the number of circles of curvature at most $X$ in a prime component ${\mathcal{P}^{\operatorname{pr}}}$, and let $C_{{\mathcal{P}^{\operatorname{th}}}}(X)$ be the corresponding quantity for a thickened prime component ${\mathcal{P}^{\operatorname{th}}}$. Let $\pi(X)$ denote the prime-counting function. Growth conjecture for prime and thickened prime components. For every prime component and thickened prime component,

$$\lim_{X\rightarrow\infty}\frac{C_{{\mathcal{P}^{\operatorname{pr}}}}(X)}{\pi(X)}=\infty,$$

and

$$\lim_{X\rightarrow\infty}\frac{C_{{\mathcal{P}^{\operatorname{th}}}}(X)}{X}=\infty.$$

The conjecture is based on experimental data; the paper proves only a lower bound for prime components and discusses estimates for the exact growth rates.