For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).

A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\mathbb{C})$ is said to be pseudo-real if there exists a matrix $C$ such that, for all $g\in SU(N)$
$$\bar{R}(g)=CR(g)C^{-1},$$
where $\bar{R}(g)$ means complex conjugation. The representation $R$ is said to be real if there exists a matrix $D$ such that $DR(g)D^{-1}$ is real for all $g$.

I would appreciate any comment or reference. Thank you!