This looks like a "lame terms" redux of Plantinga's "victorious" ontological argument from The Nature of Necessity. Here is Plantinga's explanation of why if a maximally great being exists in some possible world it exists in every possible world. He attributes the idea to Findlay (p. 214):
"Those who worship God do not think of him as a being that happens to be of surpassing excellence in this world but who in some other worlds is powerless or uninformed or of dubious moral character. We might make a distinction here between greatness and excellence; we might say that the excellence of a being in a given world W depends only upon its (non world-indexed) properties in W, while its greatness in W depends not merely upon its excellence in W, but also upon its excellence in other worlds. The limiting degree of greatness, therefore, would be enjoyed in a given world W only by a being who had maximal excellence in W and in every other possible world as well.
And now perhaps we do not need the supposition that necessary existence is a perfection; for (as I argued in Chapter VIII ) a being has no properties at all and a fortiori no excellent-making properties in a world in which it does not exist. So existence and necessary existence are not themselves perfections, but necessary conditions of perfection.
According to Plantinga himself, the argument is directed at those, who, while unconvinced that God actually exists, are willing to concede at least that his existence is possible. But why they should accept such custom made definitions of "excellence" and "greatness" is unclear. The explanation itself begins with appealing to sensibilities of those who worship God, but they presumably do not need the ontological argument to convince them of his existence.
Moreover, there is also a technical problem with making greatness in a world dependent on excellence in all other worlds. It is the same problem as with the equivocative "smallest positive integer not definable in under sixty letters" of the Berry paradox. What the integer is depends on which integers have already been defined in under sixty letters, and due to the self-referential nature of the sentence, as is, it defines nothing at all. It is the same with possible worlds, we are supposed to populate them with objects that possess certain properties. If we start making those properties dependent on what happens in all other worlds we will never finish defining even a single world, let alone all of them. This should not be surprising. The "being than which no greater can be conceived" of Anselm's original argument is already self-referential in a similar way. In mathematics such a "definition" leads to the incoherent "ordinal of all ordinals" of the Burali-Forty paradox, the ordinal than which no greater can be constructed.
See also What makes Leibniz's definition of perfection unintelligible? on problems with perfections, and Is there a suppressed premise in Anselm's Ontological Argument? on another Plantinga's reconstruction of the argument.