Option to control symbology of published point data at different scales

[AndrewBarton]

I am working with point data that I have symbolized with custom uploaded symbols. In ArcGIS Online, the only option to limit the resizing of point symbology relative to map scale is the "Adjust size automatically" option in the Symbol Style menu. Even when this option is applied, the size of my symbols still changes based on the map extent. The screenshots below show this behavior. It would be great if there was a way to keep the symbol size constant as you zoom on the map.

The second image shows the map zoomed in with the symbols shrinking in size, the third image shows the map zoomed out with the symbols growing in size: