Collapsing constant scalar curvature metrics

We prove that a sequence of constant scalar curvature (CSC) metrics which is collapsing with bounded curvature to a manifold $(X,g_{\mathrm{\infty }})$ can be perturbed to a sequence of $\mathcal{N}$-invariant collapsing CSC metrics, under a natural assumption involving the eigenvalues of the drift Laplacian on $(X,g_{\mathrm{\infty }})$. This answers a special case of a question of Cheeger-Fukaya-Gromov. We also give some natural conditions on the limiting metric-measure space under which the eigenvalue assumption is automatically satisfied.