Sorry but no there is not a way of loading center content on supporting pages. Generally it is expected that if a feature needs a supporting page then it will have its own content in the center, the support for loading the other panels is to help when that goal is to make it more seemless visually with the cms page that contains the entry point of the feature when drilling into the detail views of a content feature.
If you want to show the cms page center content it makes me think that perhaps you should not be using a supporting page at all and should just implement your feature as a cms module so it can load directly on a cms page in any panel along with other center pane content.
Supporting pages should not be that much of a replica of the functionality of the cms page. It also is bad for seo to have content duplicated on multiple urls so main content should really only be on one url. Side panels typically have ads or other things than main content that is not as problematic on multiple urls so it makes more sense to have that on supporting pages.