implications of a confirmed Lambda

imagine that future experiments have constrained the dark energy eos-parameters to be \(w_0=-1\) and \(w_a=0\) at high precision: would that imply that we have confirmed that the covariant divergence \(\nabla_\mu g^{\mu\nu}\) of the metric \(g_{\mu\nu}\) is zero, implying a metric connection, at least under the FLRW-symmetries?