The conformal method in general relativity aims at successfully parametrising the set of all initial data associated with globally hyperbolic spacetimes. One such mapping was suggested by Maxwell D (2014 Initial data in general relativity described by expansion, conformal deformation and drift (arXiv:1407.1467)). For closed manifolds, I verify that the solutions of the corresponding conformal system are stable, in the sense that they present a priori bounds under perturbations of the system’s coefficients. This result holds in dimensions 3 ⩽ n ⩽ 5, when the metric is conformally flat, the drift is small. A scalar field with suitably high potential is considered in this case.