Decomposability conjecture for D-optimal designs in the Bradley–Terry model

Let $cxi^*$ be a $D$-optimal design in the Bradley--Terry model, and let $G=([m],ctext{supp}(cxi^*))$ be its graph. A graph is decomposable if and only if it is chordal, meaning that every cycle of length four or more has a chord.

Decomposability conjecture. The graph $G$ of a $D$-optimal design in the Bradley--Terry model is decomposable.

The preceding theorem establishes that the optimal design is a rational function of the model parameter when its support graph is decomposable. Simulations indicate that $D$-optimal designs generically have decomposable graphs, but the stated assertion is not established in the supplied text.