Automorphisms of $\bar{T}$

Let $\bar{G}$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb{C}$ such that its center is trivial and $G≠\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$. Take a maximal torus $T\subset G$, and denote by $\bar{T}$ its closure in $\bar{G}$. We prove that $T$ coincides with the connected component, containing the identity element, of the group of automorphisms of the variety $\bar{T}$.