Right @tonyptony, but you’re looking at the single-cubicle model; the 4-cubicle model diagram shows an internal height and width of 12-5/8" (1/8" shy of 12-3/4"), but mine are just shy of 12-3/4" (like 12-11/16"). The 2-cubicle EKET has the same internal height as the 4-cubicle IF (and only if) positioned vertically; positioned horizontally each cubicle has an internal height of 12-1/4" (which is the width of each cubicle when positioned vertically). Confused? ;-) At any rate, the best value EKET model is the 4-cubicle version ($50 when I bought mine, now $60), and they have sufficient height for LP’s, even boxsets (though just barely. The Kallax has a little more).
With a back panel added (or even just a left-to-right brace or two), the Kallax will be great. And the underframe is made of solid wood, so should have no problem supporting a fully-loaded 4 x 4. For piece of mind you could consider adding a middle leg front and rear. A simple block of pine cut to length would do the job, or go nuts and use oak. ;-)