Let $X \subset \mathbb{R}^n$.
We say that $X$ is Conic Quadratic representable (CQr for short) if $$X=\left\{x \in \mathbb{R}^n: \exists u, A_j\begin{pmatrix}x \\ u\end{pmatrix}-b_j \in L^{m_j}, \ j=1,\dots, N \right\},$$ where $L^m=\left\{(y_1, \dots, y_{m-1}, y_m) \in\mathbb{R}^m: y_m\geq\sqrt{\sum_{i=1}^{m-1}y_i^2}\right\}$ is the Lorentz cone.
We say that $X$ is SemiDefinite representable (SDr for short) if there is an affine mapping $\mathcal{A}:\mathbb{R}^n\times \mathbb{R}^p \to \mathcal{S}^m$ such that $$X=\left\{x \in \mathbb{R}^n: \exists u \in \mathbb{R}^p, \mathcal{A}(x,u) -B \text{ is positive semidefinite}\right\},$$ where $\mathcal{S}^m(\mathbb{R})$ is the set of $m\times m$ symmetric real matrices.
I know how to prove that every CQr set is also SDr. But what is an example of a SDr set that is not CQr?
Edit: It was told to me that the collection $\mathcal{S}^m_+(\mathbb{R})$ of $m\times m$ positive semidefinite symmetric real matrices isn't CQr. But it seems hard to prove that. It is clear that it is SDr.