To evaluate davunetide 30 mg BID relative to placebo, when both are administered IN for 52 weeks, to subjects with PSP, with respect to: - Efficacy, as measured by change from baseline scores of the Progressive Supranuclear Palsy Rating Scale (PSPRS) at 52 weeks. - Efficacy, as measured by the change from baseline of the Schwab and England Activities of Daily Living Scale (SEADL) at 52 weeks. - Safety, as measured by reported adverse events (AEs), electrocardiograms (ECGs), nasal examination and clinical laboratory measures.