Gaussian Goldbach conjecture with angular restrictions

Let $n$ be an even Gaussian integer, let $N(n)$ denote its norm, and let $\arg(n)$ denote its argument. Gaussian Goldbach conjecture with angular restrictions. Every even Gaussian integer $n$ with $N(n)>10$ is the sum of two primes $n=p_1+p_2$, with

$$\arg(n)-\arg(p_j)\leq \pi/6$$

for $j=1,2$. This is a sector-restricted strengthening of the binary Goldbach conjecture in the Gaussian integers; the source presents it as a conjectural statement motivated by related work, with no resolution supplied here.