NC has a post up with the above-mentioned title. It’s a summary of a research paper and the short answer is: “Yes but it doesn’t always.”
I like this kind of research since it helps refute some propagandizing. There was a paper I read a while ago (the title of which I don’t remember), which pointed out that the other direction: “increasing prosperity => more democracy” doesn’t hold either. Quite contrary, political systems, whether democratic or not, tend to stay stable when people’s standard of living is good (or when they at least have hope that it will rise in the near future), while political changes occur otherwise.
But back to the current paper. The entire summary is interesting, and the authors show their clear-sightedness in the first section:
First, democracy may be ‘captured’ or ‘constrained’. In particular, even though democracy clearly changes the distribution of de jure power in society, policy outcomes and inequality depend not just on the de jure but also the de facto distribution of power. Acemoglu and Robinson (2008) argue that, under certain circumstances, elites who see their de jure power eroded by democratisation may sufficiently increase their investments in de facto power (e.g. via control of local law enforcement, mobilisation of non-state armed actors, lobbying, and other means of capturing the party system) in order to continue to control the political process. If so, we would not see much impact of democratisation on redistribution and inequality. Similarly, democracy may be constrained by other de jure institutions such as constitutions, conservative political parties, and judiciaries, or by de facto threats of coups, capital flight, or widespread tax evasion by the elite.