For any positive integer $n$, let $[n]:=\{1,\ldots,n\}$. Let $S_n$ denote the set of permutations (bijections) $\pi:[n]\to [n]$. For any $n>1$ and $\pi\in S_n$ we let the minimal neighbor distance be defined by $$\text{md}(\pi) = \min \big(\{ |\pi(k) - \pi(k+1)|: k\in [n-1]\}\cup \{|\pi(n)-\pi(1)|\}\big).$$ For $n>1$ denote by $E_n$ the expected value of $\text{md}(\pi)$ where $\pi$ ranges over $S_n$.

Question. Is there a positive real number $r$ such that $E_n \leq r$ for all $n>1$?

(Bonus question: What is the infimum of the values that $r$ can take?)