It seems odd that they are making a general and unbounded statement, when it is provably nonsensical.
Consider what would happen if the government tomorrow reduced taxes on the middle class and increased taxes on the wealthy to compensate. The middle class would see a rise in their disposable income. According to this paper, because taxes are now slightly more progressive, income inequality would rise. Which means that the wealthy must see an even larger rise in their disposable income. Meanwhile, government revenues remain neutral. Literally everyone wins.
We have now found a perpetual motion engine for creating wealth. We can now repeat the above process over and over again, until middle class taxes hit 0%. And literally everyone would keep getting richer in the process, without any impact on government revenues.
Clearly the above is nonsensical. There must be significant constraints on the statement that "an increase in tax progressivity actually increases income inequality"