Are there any known results of the form

there are infinitely many primes of the form $Ax^2 + By^3$

for integers $A$, $B$?

Assuming there are currently no known results of this form, what is the outlook on such problems? That is, are these problems we could plausibly attack using known methods, or do they seem to be hopelessly out of reach for now?

Commentary: I'm aware of similar results of Heath-Brown on primes of the form $x^3 + 2y^3$ and of Friedlander and Iwaniec of primes on the form $x^2 + y^4$.

On one hand, the density of numbers of the form $Ax^2 + By^3$ is larger than those of the form $x^3 + 2y^3$ or $x^2 + y^4$ ($N^{5/6}$ versus $N^{2/3}$ or $N^{3/4}$). I would naively expect this to make it easier to prove that there are infinitely many primes of the former form.

On the other hand, the proofs by Heath-Brown and Friedlander and Iwaniec apparently rely on the factorizations of the polynomials $x^3 + 2y^3$ and $x^2 + y^4$ over extensions of $\mathbb{Q}$, and $Ax^2 + By^3$ does not have such a factorization.

So it's not clear to me what I should guess about the difficulty of this problem.