Iwaniec's conjecture for primes represented by x² + y² + 1

For positive integers $N$, let $\#\{p\leq N\mid p=x^2+y^2+1\}$ count primes represented by $x^2+y^2+1$. Define the constant

$$C_1:=\lim_{N\to\infty}\frac{\#\left\lbrace p\leq N\;\middle|\;p=x^2+y^2+1\right\rbrace}{N\log^{3/2}N}.$$

Iwaniec's conjecture. The limit exists and equals

$$C_1=\frac{1}{\sqrt{2}}\prod_{p\equiv3\bmod4}\left(1-\frac{1}{p^2}\right)^{-1/2}\left(1-\frac{1}{p(p-1)}\right)\approx0.610534\dots.$$

This is an amended version of a conjecture attributed in the source to Motohashi. The paper presents numerical evidence, but the asserted asymptotic remains conjectural.