Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor defined in the reference) is conjectured to represent primes with density $\Gamma_G/\log x$. (What is the name of this conjecture? Is it Bouniakowsky? Or Hardy-Littlewood's F?).
This conjecture is wide-open. In fact, according to MathWorld, it seems that it is not even known whether the density is non-zero: such polynomials, while expected to represent primes infinitely often, are not proven to represent primes at least once.
If we strengthen the hypotheses, such as $g\equiv2$ (and, say, negative discriminant), is there any known result concerning the density of primes? Is it known whether quadratic polynomials of the form above represent primes at least once?