Inflation isn't caused by decreasing the wage gap. Inflation is caused by monetary policy. The two are orthogonal, so there is no answer, because it's like asking to explain the effects of a NYC pizzeria changing its box colors on the average food consumption levels of animals in the San Diego Zoo. Sure, it could be studied, but is unrelated, so nobody has.
There is no link between minimum wage and inflation, so explaining the link is impossible.
Oh, and in countries with a high minimum wage, the people on minimum wage make more than America's middle class, so a higher wage for those people would provably improve the middle class. When the minimum wage is set well below poverty level, the effect isn't the same. Set the minimum wage at $25/h or higher, define "middle class" as $50k to $60k, and move the minimum wage from $25 to $30 and tell me the effects on the middle class.