Do rising inequalities create inflation?
This is not an answer that will state exactly which assumptions are valid/invalid but rather a guide to find answers to your question.

Here are various scientific articles about the topic:

These first two papers take what is called a Political Economy perspective on the matter. They assume that the central bank controls inflation, but the central bank is to some extent controlled by the population. If the population has higher income inequality, it prefers higher inflation and thus countries with higher inequality will face higher inflation.

Albanesi, Stefania. "Inflation and inequality." Journal of Monetary Economics 54.4 (2007): 1088-1114.

Dolmas, Jim, Gregory W. Huffman, and Mark A. Wynne. "Inequality, inflation, and central bank independence." Canadian Journal of Economics/Revue canadienne d'économique 33.1 (2000): 271-287.

This paper is also interesting, since it considers the different inflation faced by different income classes. It is somewhat orthogonal to your question, however.

Hobijn, Bart, and David Lagakos. "Inflation inequality in the United States." review of income and Wealth 51.4 (2005): 581-606.

However, to my knowledge, there are no papers written on income inequality as a direct cause of inflation without the involvement of the Political Economy argument above. This is mainly because in Economics our models usually tell us that the central bank has almost perfect control over inflation (except for negative inflation rates).

PS: If you find the readings above too difficult, consider reading a textbook on monetary economics:
Mishkin, Frederic S. The economics of money, banking, and financial markets. Pearson education, 2007.