Replacing a Regressor With its Conditional Expectation

#1
Consider a case of measurement error, where Y=Xβ+ϵ but we only observe Z=X+u. I know the standard approach to this is to use IV. What if we don't have an instrument, but we do know (read: assume) the distribution of X and u, so we can derive the conditional expectation E[X|Z]. If we replace Z with an estimate of E[X|Z], can we get a consistent estimate of β?