Bounded odd Gaussian-prime representation conjecture

Let $\Gamma_G$ denote the Gaussian integers under consideration in the source, and let

$$K_\pi=\{z\in\pi_{\mathbb{Z}[i]}\mid\operatorname{Re}(z)\geq0\text{ and }\operatorname{Im}(z)\geq0\}.$$

Bounded odd Gaussian-prime representation conjecture. There are constants $c_0$ and $k_0$ such that, for every $z\in\Gamma_G$ with $\max(\operatorname{Re}(z),\operatorname{Im}(z))\geq c_0$, $z$ is a sum of at most $k_0$ odd primes from $K_\pi$. This is formulated using the preceding theorem and an integer additive hypothesis; it remains open in the source.