Stability of determination of Riemann surface from its DN-map in terms of Teichmüller distance

As is known, the Dirichlet-to-Neumann operator $\mathrm{\Lambda }$ of a Riemannian surface $(M,g)$ determines the surface up to conformal equivalence class $[(M,g)]$. Such classes constitute the Teichmüller space with the distance ${\mathrm{d}}_T$. We show that the determination is continuous: $\mathrm{\parallel }\mathrm{\Lambda }-{\mathrm{\Lambda }}^{\mathrm{\prime }}{\mathrm{\parallel }}_{H^1(\mathrm{\partial }M)\to L_2(\mathrm{\partial }M)}\to 0$ implies ${\mathrm{d}}_T([(M,g)],[(M^{\mathrm{\prime }},g^{\mathrm{\prime }})])\to 0$.