The ontological argument

peace,
I have a question regarding the ontological argument.
First, I will briefly describe the argument in the initial stage as I understand it.
Assumption: 1. Even an atheist can conceive of the greatest being.
2. Part of the measure of an object's greatness is its (necessary) existence or non-existence.
Conclusion: The greatest being must exist (and even must)
There is seemingly a possible solution to the ontological argument. As soon as an atheist conceives of the greatest being because he proceeds from the fundamental assumption that God does not exist, he excludes the measure of existence from the measure of the greatness of the greatest being.
I am trying to attack mainly the application of assumption number 2 in the matter of assumption number 1. Is there a logical fallacy in the explanation I proposed?