The Carathéodory metric on Teichmüller space of genus two surface

Let ${\text{Tei}}_{g,n}$ be the Teichmüller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichmüller and Carathéodory metrics agree on a Teichmüller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for ${\text{Tei}}_{0,5}\cong {\text{Tei}}_{1,2}$. We confirm the conjecture for ${\text{Tei}}_{2,0}\cong {\text{Tei}}_{0,6}$.