I have a die that produces uniformly distributed values in $\{1,\ldots, k\}$ for some integer $k\geq 2$. Now I play the following game.

I start rolling the die and produce one integer in $\{1,\ldots,k\}$ after another, $X_1,X_2,\ldots$, and I stop when my most recent integer $x$ lies between the previous integer $b$, and the integer $a$ obtained before that. (More formally, the stopping time is the least $n\ge 3$ such that $X_n\in [\,\min\{X_{n-2},X_{n-1}\}, \max\{X_{n-2},X_{n-1}\}\,]$.)

An example: $k=6$, and my dice rolling sequence is $2, 4, 5, 1, 4$ $\implies$ there I stop, since $4$ is in the interval $[1,5]$ of the last two dice rolls $5$ and $1$.

Let $E_k$ be the expected value of the length of one game with a dice of $k$ sides. It may be difficult to give an explicit value for $E_k$, but:

**Question.** Is $\{E_k:k\in\mathbb{N}, k\geq 2\}$ bounded?