Rationality conjecture for fully supported Bradley–Terry optimal-design weights

Let $m$ be the number of alternatives, let $\binom{m}{2}$ pairwise-comparison intensities be denoted by $\boldsymbol{\lambda}=(\lambda_{ij})$, and let $w_{ij}$ be the weights of a fully supported $D$-optimal design. Rationality conjecture. The $\binom{m}{2}$ weights are rational functions in the intensities, with numerator degree

$$\binom{m}{2}+m-1.$$

The conjecture predicts an explicit algebraic form for the weights obtained from the optimality equations; the paper gives degree-$9$ expressions in the four-alternative case, while the general claim is left open.