Engineering help needed

That shelving is probably made of pressboard with a wood veneer finish. At times these manufactured boards only have a veneer on the top and front edge of the board (the areas that are seen). But occasionally the removable shelves will have the finish on all sides. In those cases the boards could simply be flipped to put the bow on the top. It would eventually level out and then return to the state it is now after some time (if the weight remains the same).

But mjcouvi has the right idea to install some 'L' brackets under each sagging board vertically down the center of your shelving case. If you purchase 5 of these brackets, one for each sagging horizontal board (excluding the bottom of course), you can attach these to the back panel where they will support the shelves in the middle. You will need to ensure that the top of each bracket is on a horizontal plane that is level with the top of the pegs that you intend to use so that these middle supports are at the same height as the pegs. And since they are merely supports for the shelves, you shouldn't even have to put screws on the top side of the brackets; just screw on the vertical leg that attaches to the back panel.

'L' brackets come in many styles and sizes and it may be possible to find a type that can be installed without even being noticeable once the items are put on the shelves. I'd be interested in seeing what you come up with and how it looks when completed.