Full support of the zero-temperature Sherrington-Kirkpatrick order parameter

We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0,1)$. Thus its support has no gaps at any overlap scale below one. We use the covariance normalization $\xi (t)=t^2/2$.