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Dramatic_Hammer
If you are VERY careful with their placement then french cleats would work well at supporting that with the added bonus (assuming you only want to have a shelf for support) that you could dispense with the supporting shelf. One fixed to the bottom rail of the frame to support the base and one fixed to each upright about 2/3 of the way up.
https://youtu.be/KXHNvNshlSE
Alternatively these floating shelf brackets are good. Easy to install and strong.