  The aim of this paper is to show that homotopy pro-nilpotent structured ring
spectra are TQ-local, where structured ring spectra are described as algebras
over a spectral operad O. Here, TQ is short for topological Quillen homology,
which is weakly equivalent to O-algebra stabilization. An O-algebra is called
homotopy pro-nilpotent if it is equivalent to a limit of nilpotent O-algebras.
Our result provides new positive evidence to a conjecture by Francis-Gaisgory
on Koszul duality for general operads. As an application, we simultaneously
extend the previously known 0-connected and nilpotent TQ-Whitehead theorems to
a homotopy pro-nilpotent TQ-Whitehead theorem.
