Attach damped folding doors to Ikea Besta bench

Hello, I bought two Besta benches (120x49x26) as a lowboard. Unfortunately, for this Besta bench there are only drawers and no flap doors. I want to attach the drawer fronts so that I can open them from top to bottom. Sounds simple... Cup hinges have already been obtained, but I can't find suitable rails anywhere, neither at the hardware store nor at the specialist dealer, to prevent the flaps from simply dropping down unbraked when opened. That means I urgently need appropriate rails to attach to the sides that slow down or dampen the opening of the flap. The problem is that the compartments of the Besta bench are only about 19-20 cm high and I cannot find braking rails for this, as all those I have found so far require a height of at least 25 cm. I have also often seen round guide rails for flap doors in cabinets/shelves, which run through a plastic part that is then attached to both sides of the compartment. That would also work. Who can please help me? Is there maybe a suitable solution from Ikea? Or other suggestions? Thank you very much!

Have a look at Blum.at. There, everything important about hinges is listed. Also damping systems, etc.