Almost-primes in Sun's $x^2+n{y}^2$ conjecture

In 2015 Zhi-Wei Sun proposed the conjecture that any integer $n>1$ admits a partition $n=x+y$ with integers $x,y>0$ such that $x+ny$ and $x^2+n{y}^2$ are simultaneously prime. To approach this conjecture we use the method of weighted sieve as developed by Richert, Halberstam, and Diamond. In this article, we first formalize the conjecture into a sieve problem. We verify that the conditions required to use Richert’s weighted sieve are satisfied and establish partial results with almost-prime solutions for sufficiently large $n$.