For a normal production site, leave Auto Synchronize off, which is the default.
Why you usually do not need it. Your site already stays current automatically. HomeRunner Central pushes updates to your properties in real time as they change, new properties are imported every hour, and properties removed in your PMS are cleaned up every hour. None of this depends on the Auto Synchronize toggle.
What Auto Synchronize adds. It adds one thing: a scheduled full re-sync of every existing property. It is the right tool in a few situations: a staging or pre-launch site that Central’s live updates cannot reach yet; a site that needs fields such as photos, house rules, or custom fields refreshed on a schedule, since those fields are not part of the real-time update set; or temporarily repairing data drift after an outage.
If you enable it. Set the interval to several hours or nightly. A full sync on a large catalog can take about two hours, and the interval should always be longer than the sync itself. Keep Skip Auto Sync Rescheduling enabled so that overlapping runs are skipped instead of piling up.