Power-logarithmic asymptotic conjecture for Tate–Shafarevich group orders in the Neumann–Setzer family

For $i\in\{1,2\}$, let $f_k(i,X)$ count integers $u\equiv1\pmod 4$ with $|u|\leq X$ satisfying the paper's condition $(**)$, such that $L(E(u),1)\neq0$ and $|$\cyrfont X$(E_i(u))|=k^2$. Asymptotic conjecture for the Neumann–Setzer family. For every positive integer $k$, there are constants $c_{k,i}>0$, $\alpha_{k,i}$, and $\beta_{k,i}$ such that

$$f_k(i,X)\sim c_{k,i}X^{\alpha_{k,i}}(\log X)^{\beta_{k,i}},\qquad X\to\infty.$$

This is a numerical conjecture about the frequency of prescribed square orders of Tate–Shafarevich groups in the family; no resolution is given in the source.