Local deformation conjecture for arithmetic isometric actions of complex hyperbolic lattices

Let $\rho$ be an arithmetic isometric action of a lattice in $SU(1,n)$. Arithmetic-action deformation conjecture. Any sufficiently small perturbation of $\rho$ is of the form obtained by deforming along the centralizer construction described immediately before the conjecture. The claim is an infinite-dimensional analogue of deformation-rigidity results of Goldman--Millson and Corlette, but the source gives no resolution.