Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let 
\begin{equation}
f: \mathcal{X} \rightarrow \mathbb{R}, \quad g: \mathcal{X} \rightarrow \mathbb{R}^{1\times m},\quad h: \mathcal{X} \rightarrow \mathbb{R}_+, \quad S: \mathcal{X} \rightarrow \mathbb{R}^{m\times m}
\end{equation}
be continuously differentiable functions and assume $S(x)$ corresponds to a strictly positive definite matrix for every $x\in\mathcal{X}$.

Consider the correspondence $\Pi :\mathcal{X}\rightarrow 2^{\mathcal{P}}$, given by
\begin{equation}
\Pi(x) = \left\{p \in \mathcal{P} \ \Big\vert  \ f(x) + g(x)p \geq \sqrt{h(x) + p^\top S(x) p \ \ } \ \right\},
\end{equation}
and assume that $\Pi(x)$ is non-empty for every $x\in\mathcal{X}$.

$\textbf{Question: Is the correspondence }$ $\Pi$ $\textbf{Hausdorff-Lipschitz in}$ $\mathcal{X}$, i.e., does there exist a positive scalar $L$, such that
\begin{equation}
d_H( \Pi(x), \Pi(x')) \leq L\Vert x-x'\Vert_2
\end{equation}
hold for every $x, x'\in \mathcal{X}$, where $d_H$ denotes the Hausdorff distance?