The relative-density conjecture for primitive smooth solutions

Let $N^*(H,\kappa)$ denote the number of primitive smooth solutions and let $N(H,\kappa)$ denote the total number of smooth solutions below $H$. For $1<\kappa<\infty$, consider the relative density $N^*(H,\kappa)/N(H,\kappa)$ as $H\to\infty$.

Relative-density conjecture for primitive solutions.

$$\lim_{H\to\infty}\frac{N^*(H,\kappa)}{N(H,\kappa)}= \begin{cases} \dfrac{1}{\zeta\left(2-\dfrac{3}{\kappa}\right)},&3<\kappa<\infty,\\ 0,&1<\kappa\leq 3. \end{cases}$$

The conjecture predicts a qualitative transition at $\kappa=3$: primitive smooth solutions have positive limiting relative density above the threshold and density zero at or below it. The source presents this as conjectural and gives no proof or disproof.