There’s really two parts to this question. The first is what happens to those who die in original sin only with no actual sins (ie if they die in original sin and God does not cleanse them of it)? Of course, the only people this could really apply to would be infants. The second question is, is there a possibility that infants who are not baptized (and cannot seek it), that these are cleansed of original sin and then saved in some other way? (ie is it possible they do not die in original sin?).
For the first question, it is dogmatic (even to now) that those who actually die in original sin do not enter Heaven (they do not see God, partake of His nature, however you want to describe it). There seems to be a split in opinion among the Fathers ranging between this being a state of natural happiness (what we call Limbo) with the only “punishment” being exclusion from Heaven, to them suffering the actual tormenting punishments of Hell. This latter opinion mostly died out long ago (resurrected briefly by the Jansenists), but it has never been definitively condemned. The dogmatic definitions of the Church on this topic only say the punishments are different between original sin and actual sins.
For the second question, there has also been different opinions, with the more common teaching for a long time being it was unlikely these were cleansed (or at least, this was the prudent position to take, since God did not reveal a manner of salvation for them). Those that proposed otherwise, proposed possibilities ranging from a gratuitous gift of God to the vicarious faith of the parents or the Church providing the means. Nowadays, these latter opinions, especially that of the gratuitous gift, are the more common. But still, the Catechism, does not say definitively one way or the other, just that we can pray for (as we do in our rites) and hope for it.
I think both East and West agree that God has simply not revealed it to us which is why even we in the West cannot provide a definitive answer. And so there has been a range of approaches and proposed answers in both schools, moreso of course in the West. But, from what I can tell, while we have both gone through periods where we had a more pessimistic view, now we are in general agreement in having a bit more optimism overall.