As that paper very first assumption is both unlikely and shows a complete lack of imagination, I think its results are dubious. In addition, I seriously doubt that we hae the math to be able to predict this.

Here is what the paper says:

Let us start with the worst-case scenario (for overpopulation)—physical immortality... Consider a situation, when parents produce less than 2 children on average, so that each next generation is smaller than the previous one... This conclusion does not require any complex calculations and questionable assumptions, but follows directly from the calculus, and the property of infinite geometric series to converge

Well, I beg to differ with the "questionable assumption" part. Why would anyone think that in infinite time, parents will have, on average, less than two children?

And here's the lack of imagination: the whole (cultural) concept of "parents" and family dissolves when there's no death. If you marry at 1000, and stay married to your wife for 5000 years (producing one child), chances are that by the time you're 50000 you've likely forgotten your wife's name, and your only child's too. In fact, you won't even remember that happened (well, maybe you'll remember your first time around, but by your 200th marriage, you won't remember how many children you had with your 37th wife).

Of course, I may be suffering from lack of imagination, and the structure of society will be completely different. The point is that making any assumption on human behavior after such a revolution of much of what makes us human, is preposterous. Immortal humans will behave in ways that we cannot predict, and we therefore cannot predict how the population will grow.