Conjecture: Let $\mu_x$ be the arithmetic mean of the ratio of the perimeter to the hypotenuse of all primitive Pythagorean triplets in which no side exceeds $x$; then,$$ \lim_{x \to \infty}\mu_x = 1 + \frac{4}{\pi}$$

Based on data for $x \le 10^{11}$, the computed value agrees with the conjectured value up to $5$ decimal places.

Primitive Pythagorean triplets $a^2 = b^2 + c^2, \gcd(b,c) = 1$ are given by $a = r^2 + s^2$, $b = r^2 - s^2$ and $c = 2rs$ where $r > s$ are natural numbers. Let the $n$-th primitive triplet be the one formed by the $n$-th smallest pair in increasing order of $(r,s)$. It was proved in this post MSE that the arithmetic mean $\mu_n$ of the ratio of the perimeter to the hypotenuse of first $n$ primitive Pythagorean triplets approaches $\dfrac{\pi}{2} + \log 2$ as $n \to \infty$. However the above claim is still open.

**Question**: I am looking for a proof or disproof of this conjecture.