I doubt it.
The whole point of moving shop is avoiding the consequences of the choices made in ME3. That's only possible if we leave before. That doesn't mean, however, that we're leaving with a small population (five hundred thousand, five million, fifty million... all those are negligible portions of a galaxy wide demographic) and the game doesn't have to take place immediately after we arrive.
The thing though, is how do we know that? Most of that belief is conjecture and just fans theorizing on the timetable.
The Mass Effect 4 leak said the following if you remember:
The next Mass Effect game takes place in the Helius Cluster (a cluster of 100s of solar systems in the Andromeda Galaxy), far removed by time and space from Commander Shepard’s heroic acts and the final events of the Mass Effect trilogy
The thing that sticks out from me is how it says far removed by time and space. Yes by distance Andromeda is far from the Milky Way, we all agree on that. But time and space also implies it takes place after the trilogy events.
The wording is ambiguous at worst, but if you ask me, time and space is basically saying years, centuries, whatever it is, from the events of the trilogy. If the events were during Mass Effect 3 before the final battle, there be no reason to mention "far removed by time and space", they would literally say before the final battle takes place.
Not to mention much of the writeup would also be different. The Pathfinder Initiative seems like a sort of "Lewis and Clark" type of group tasked for exploration over preservation, but thats how I read into it.
The real issue in the end is we just don't know until they tell us.