mean displacement inequality for random walk with drift away from zero

Question

Suppose $X_n$ is a nearest neighbor random walk on the integers with transition probabilities biased towards moving away from zero but with the bias asymptotically vanishing as you move away from zero. Specifically, for $x\in\mathbb{Z}$, we have the probability of moving to the right $P(X_{n+1}=x+1|X_n=x)=\frac{1}{2}+\frac{x}{x^2+1}$ and of moving to the left $P(X_{n+1}=x-1|X_n=x)=\frac{1}{2}-\frac{x}{x^2+1}$. It seems intuitively obvious that $E_x|X_n-x|\leq E_0|X_n|$ for all $x\in \mathbb{Z}$ and $n\geq 0$ but I've been unable to come up with a proof. Showing $E_x[(X_n-x)^2]\leq E_0[X_n^2]$ for all $x\in \mathbb{Z}$ and $n\geq 0$ would also be sufficient for my purposes as would the appearance of a constant independent of $n$ but that doesn't seem to make the problem any easier. Hints and references are welcome.