Permutations from ranking independent random variables

Let $X_1,\ldots,X_n$ be almost surely distinct independent real-valued random variables. Let $σ$ be the random permutation of $\{1,\ldots,n\}$ such that $X_{σ(1)}<X_{σ(2)}<\cdots< X_{σ(n)}$. We show that the set of laws of $σ$, as the laws of $X_1,\ldots,X_n$ vary, has semialgebraic dimension \[