Let $\mathcal M_X(r,0,K_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$. Then clearly we have an injection $$H^0(X,E\otimes E^*\otimes K_X)\rightarrow \mathcal M_X(r,0,K_X)$$

Is this map a closed immersion? Is it proper?

Thanks