Prime values of $n^2+1$ conjecture

Let

$$C_+=2\prod_{p>2}\left(1-\frac{\left(\frac{-1}{p}\right)}{(p-1)^2}\right)=1.3728134628\ldots,$$

where $\left(\frac{-1}{p}\right)$ is the Legendre symbol. $n^2+1$ prime-value conjecture. The expected number of primes of the form $n^2+1$ less than or equal to $N$ is

$$\frac{C_+}{2}\int_2^{\sqrt N}\frac{dx}{\log x}\sim C_+\frac{\sqrt N}{\log N}.$$

This is the local-density heuristic for the quadratic polynomial $n^2+1$; it remains open.