On a Gauss inequality for the unimodal distributions

Let $\xi$ be a random variable with an unimodal distribution, $M$ be a mode of this distribution, $x_0\in(-\infty,\infty)$ and $\theta^2=\mathbf D\xi+(\mathbf E\xi-x_0)^2=\mathbf E(\xi-x_0)^2$. It is shown that for all $k\ge 2$
$$ \mathbf P\{|\xi-x_0|\ge k\theta\}\le\frac{4}{9k^2}. $$
if the point $x_0$ separates the points $M$ and $\mathbf E\xi$ then the inequality is fulfilled for all $k\ge\sqrt 3$.