Which pairs should you compare? An Evidence Press briefing. Many rankings are built from head-to-head comparisons: chess ratings, taste tests, and votes between two answers from language models. The Bradley–Terry model gives each item a strength, and one item beats another with a probability given by its share of their combined strength. If you have a fixed budget of comparisons, you must decide how much of it goes to each pair. The best choice, in a precise statistical sense, is called a D-optimal design: it makes the joint estimate of all the strengths as sharp as possible. The pairs that such a design actually uses form a network. In 2025, Frank Röttger, Thomas Kahle and Rainer Schwabe conjectured that this network is always chordal: every loop of four or more items has a shortcut across it. That property gives their exact formulas for optimal designs. This paper shows the conjecture is false. With fourteen items, six weak, six strong and two in between, the unique best design uses fifty-four of the ninety-one possible pairs, and those pairs contain a loop of four with no shortcut. Its weights are not even rational numbers. Every one of the ninety-one optimality conditions was checked in exact arithmetic. The effect is not an accident of tied strengths: it survives small changes, appears for every number of items from fourteen upwards, and has an example with all strengths different. Some structure does survive: a loop of four or more can never stand alone as a separate block of the network. In practice the loss is small: a chordal design comes within about a sixth of one per cent of the best. The lesson is for people who compute exact optimal designs. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or formally verified. Published on the third of November twenty twenty-six, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.