Nope. Marginal propensity to spend kills that out of the gate. Poor people save a much smaller proportion of their income than rich people, on average. Since rich people are rich (meaning that percentage is out of a bigger pie than the poor person's),
and their savings rates are higher than poor people, you just multiply that out and see that rich people will get richer much faster than poor people.
This doesn't take into account taxes, however, which means that we're assuming a 0% flat tax rate. But, you see, what happens is that if you start imposing a positive taxation rate, then all that changes in the preceding argument is the definition of who is "rich" and who is "poor." The general trend is all that matters.
One actual limitation to point out is that this argument doesn't take investment into account. Over long enough time periods, all major asset classes have a positive expected return, so, it doesn't matter precisely what people invest in. What does matter is how much they have to invest, and how long "long enough" time periods are.
IIRC, for the stock market, "long enough" means 15 years or so. Since this is much shorter than a single human lifetime, you see that all someone who is rich enough to invest has to do is just keep shovelling money into their brokerage account, and it will pay off eventually.
Poor people don't even get to that point. The hows and whys of that are myriad, so I won't even get into that. Just imagine that, instead of calling them "poor people," we call them "economically fragile" people. Then, what you start to realize is that a small bump in the road, like, say, an unexpected car repair, can really devastate an economically fragile person's finances.
And, because economically fragile people don't have a lot of money coming in, their expenses are pretty low. That means that even if they follow standard personal finance advice and try to accumulate a 6 month emergency fund, a car repair costs what a car repair costs, so it's going to eat a much larger chunk of that emergency fund for the economically fragile person than the non-economically fragile person.
Oh, and, because economic fragility is relative (meaning if you, and everybody else have 1 quatloo each and I have 10 quatloos, I'm still as "rich" overall as if everybody had 10 quatloos each, and I had 100), that means starting poor and getting rich takes a long, long time, and may not ever happen.
TL;DR: A completely flat tax rate for all citizens implies that the rich will get richer, and the poor will stay fairly poor.
https://en.wikipedia.org/wiki/Marginal_propensity_to_consume