The Eisenstein Ghost Twins

As always in this Goldbach setup for Eisenstein primes, we take w as the cube root of -1 in the first quadrant. It is w=(1+sqrt(3))/2. Positive Eisenstein integers are Eisenstein Integers of the form a+wb with positive a and b. The are located in the positive first sextant sector of the complex plane.

The Eisenstein Ghost Twins are examples of positive Eisenstein integers a+wb with a=3, b>2 or b=3,a>2 which can not be written as a sum of two positive Eisenstein integers. It appears that every Eisenstein integer a+wb with a=2 or b=2 can be written as a sum of two positive Eisenstein Primes. These two (unique?) counter examples are the reason, why the Eisenstein Goldbach conjecture is formulated as
"Every positive Eisenstein integer a+wb with a and b larger than 3 is a sum of two positive Eisenstein primes".
It pairs with the Gaussian Prime Conjecture: "Every even positive Gaussian integer a+i b with a and b larger than 1 is a sum of to positive
Gaussian primes.
But it is the existence of the singular Eisenstein ghosts which prevent stating the conjecture for a,b larger than 1 also in the Eisenstein case.