Primes of the form l³ + m³ + k

Let $k$ be a fixed nonzero integer, and let $P(n)$ be the number of primes of the form $l^3 + m^3 + k$ less than $n$, with $l, m$ positive integers, each prime counted as often as it is so represented. Let $\rho = e^{2\pi i/3}$. For a prime $q \equiv 1 \pmod 3$, let $\omega = u + v\rho$ be the prime factor of $q$ in $\mathbb Z[\rho]$ with $u \equiv -1$ and $v \equiv 0 \pmod 3$, and define $A, B$ by $A = 2u - v$, $3B = v$, so that $4q = A^2 + 27B^2$. For such $q$ not dividing $k$, let $$ A_q = -\frac{A - 2}{q(q-1)} $$ if $-k$ is a cubic residue of $q$, and $$ A_q = \frac{\tfrac12 A \pm \tfrac92 B - 2}{q(q-1)} $$ otherwise, with the plus sign if the cubic residue symbol $\bigl(\frac{-k}{\omega}\bigr)_3$ equals $\rho$ and the minus sign otherwise.

Problem. Prove that there are infinitely many primes of the form $l^3 + m^3 + k$ with $l, m$ positive, and that $$ P(n) \sim \frac{\Gamma(4/3)^2}{\Gamma(5/3)} \frac{n^{2/3}}{\log n} \prod_{p} \left(1 - \frac{2}{p}\right) \prod_{q} (1 + A_q) \qquad (n \to \infty), $$ where $p$ runs over the primes $p \equiv 1 \pmod 3$ dividing $k$, and $q$ over the primes $q \equiv 1 \pmod 3$ not dividing $k$.