Stevenhagen's negative Pell equation density conjecture

Let $D$ range over fundamental positive discriminants without prime factors congruent to $3$ modulo $4$. Define

$$\rho:=\prod_{j=1}^\infty \left(1+2^{-j}\right)^{-1}=0.41942\cdots.$$

Stevenhagen's conjecture. The density of those $D$ for which the negative Pell equation

$$x^2-Dy^2=-4$$

has a solution is $1-\rho$. This conjecture predicts the proportion of such discriminants for which the associated negative Pell equation is solvable; its stated density is a central quantitative result in the arithmetic of quadratic fields and has been resolved.