Anick's approximation conjecture for elliptic spaces

Let $S$ be a simply connected finite CW complex, let $n$ be a natural number, and let $S$ be a simply connected finite Postnikov piece when the second formulation is considered. An $n$-equivalence is the corresponding approximation map through degree $n$. Anick's approximation conjecture. Any simply connected finite CW complex $S$ can be approximated arbitrarily closely on the right by an elliptic space: for each natural number $n$ there is an elliptic space $E_n$ and an $n$-equivalence $S\to E_n$. Equivalently, any simply connected finite Postnikov piece $S$ can be approximated arbitrarily closely on the left by an elliptic space: for each natural number $n$ there is an elliptic space $E_n$ and an $n$-equivalence $E_n\to S$. The source attributes this problem to Anick and cites its formulation by Félix--Jessup--Murillo; no resolution is stated in the supplied text.