So this is what I did.
Added screws from the back for each shelf:
Screwed a 2mm thick and 25mm wide Aluminum strip to the front of each shelf. I also put tension on the Aluminum so it would counteract the sagging (pressed downwards as hard as I could on the middle part while putting in the screws).
I got them from my scrap metal heap, else i would have ...
As clarified in the Comments, the back panel is thick enough MDF that fastening in through from the back is viable and that alone will do quite a bit to resist any tendency towards sagging.
If you want to stiffen the shelves still further there are various established ways of doing this for plywood that could be adopted here, e.g. the below:
But the ...