We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body ¿¿Rn, n¿3, by means of the so-called local Neumann-to-Dirichlet map on a curved portion ¿ of its boundary ¿¿. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in Inverse Problems 33 (2018), 125013, we provide a Hölder stability estimate on ¿ when the conductivity is a-priori known to be a constant matrix near ¿.