Three odd Gaussian primes conjecture for the first quadrant

Let $A=\{z\in\mathbb{Z}[i]\mid\operatorname{Re}(z)>0\text{ and }\operatorname{Im}(z)>0\}$, and let $K_\pi=\{z\in\pi_{\mathbb{Z}[i]}\mid\operatorname{Re}(z)\geq0\text{ and }\operatorname{Im}(z)\geq0\}$. Three odd Gaussian primes conjecture. For every $z\in A$ with $\max(\operatorname{Re}(z),\operatorname{Im}(z))\geq7$, the Gaussian integer $z$ is a sum of at most three odd primes from $K_\pi$. This is motivated by computational experiments on representations of Gaussian integers; the source gives no proof or resolution.