The quadratic-prime conjecture for $n^2+1$

Let $f(n)=n^2+1$, and define its weighted prime-value counting function by

$$S_2(x)=\sum_{n^2+1\leq x}\Lambda(n^2+1).$$

Quadratic-prime conjecture. There are infinitely many primes of the form $p=n^2+1$, and

$$S_2(x)=s_2x^{1/2}+O\left(\frac{x^{1/2}}{\log x}\right).$$

This is a special case of the prime-values heuristic and is open; in particular, infinitude of primes represented by $n^2+1$ is not known.