Stationary Distribution of a Markov Process defined on the space of permutations

Let $ S$ be the set of $ n!$ permutations of the first $ n$ integers. Let $ p\in(0,1)$ . Consider the Markov Process defined on the elements of $ S$ .

1. Let $ x\in S$ . Choose two distinct integers $ 1\le i <j \le n$ uniformly at random among the $ n(n+1)/2$ possible combinations.
2. If $ x_i < x_j$ , swap $ x_i$ and $ x_j$ with probability $ p$ , otherwise do nothing. If $ x_i > x_j$ , swap $ x_i$ and $ x_j$ with probability $ 1-p$ , otherwise do nothing.

This process is ergodic, because there is path between any two states with non-zero probability. It has a stationary distribution. I conjecture that the stationary distribution of $ p(x)$ depends only on $ p$ and on the number of mis-rankings of $ x$ , defined as $ \sum_{i\le j} 1\{x_i < x_j\}$ . But am not able to prove it. I also wonder whether this simple model has been studied somewhere, maybe in Statistical Mechanics. Any literature reference is appreciated.

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