I don’t know about the Kallax, but I have my Ikea EKET cubes sitting on the underframe Ikea made (now discontinued, replaced by individual feet) for it. The 4-cube EKET is 27.5" wide, and I have three EKET’s on each underframe, so 12 cubicles total on each frame
The EKET has holes drilled into the back ends of each (top, bottom, and both sides), made for installing brackets (sold as an accessory option) that secure one cube to another. My stacks of EKET’s are so secured, as well as to EKET’s on either side---three 4-cube EKET’s tall, five wide. The EKET also comes with adjustable spacers that are attached to their backside, for stabilizing each to the wall behind them. Each underframe foot (and the individual feet) is also adjustable, so I have the stack of EKET’s leaning just slightly towards the wall, the adjustable back spacers making contact with that wall. My assembly of fifteen 4-cube EKET’s feels like it’s part of the room’s structure; absolutely immovable!
What you might consider is adding another leg in the middle of the Kallax underframe, front and rear. That will of course provide support in the middle of each underframe, the weak point in the frame. If you have yet to assemble the Kallax’s, take a look at the EKET. I compared the two on the sales floor, and found the EKET to be more structurally stiff than the Kallax, with the bonus of each cubicle being an enclosed space. The EKET has a back panel, the Kallax does not. With an open back side, dust can get to the open side of your LP’s. If you have your LP’s in plastic outer sleeves, perhaps that’s not a problem. Another EKET advantage is the shallower depth; the EKET is only 13.75" deep, so doesn’t have the excessive space behind the LP’s that the Kallax does. In the EKET, the LP’s come right to the front edge of the cube.