Let $k$ be an algebraically closed field and let $K/k$ be a field extension of infinite transcendence degree where $K$ is algebraically closed. Is it true that $\mathrm{Aut}_k(K)$ is a simple group?

The above question is motivated by work of Lascar. Lascar gives a proof of the above statement when $k=\overline{\mathbb{Q}}$ and $K=\mathbb{C}$ in his article The group of automorphisms of the field of complex numbers leaving fixed the algebraic numbers is simple.

Automorphism groups of fields, and their representationsby Rovinskii, according to which Lascar actually proves the result for any extension of algebraically closed fields of uncountable transcendence degree. – François Brunault May 12 '13 at 18:11