The asymptotic probability of being an a-Sophie Germain prime

Let $p$ be prime and let $a$ be a parameter for which the paper defines the notion of an $a$-Sophie Germain prime (or $a$-SGP). Let $\alpha_3(a,n)$, $\alpha_4(a,n)$, and $\alpha_5(a,n)$ denote the three empirical quantities used in the paper.

The a-Sophie Germain prime probability conjecture. The probability that a given prime $p$ is $a$-SGP converges, as $p\to\infty$, to the common asymptotic value represented by

$$\lim_{n\to\infty}\alpha_3(a,n)\approx\lim_{n\to\infty}\alpha_4(a,n)\approx\lim_{n\to\infty}\alpha_5(a,n).$$

The claim is based on numerical computations for several values of $a$ and is presented without a proof; its status is therefore open in the supplied source.