Vallentin's conjecture for the least Euclidean distortion of distance-regular graphs

Let $G$) be a distance-regular graph with diameter $d$, valency $k$, eigenvalues $\theta_0>\theta_1>ots>\theta_d$, and standard sequences $(w_0(\theta_j),...,w_d(\theta_j))$ corresponding to $\theta_j$. Let $c_2(G)$ denote the least distortion required to embed the shortest-path metric of $G$ into Euclidean space. Vallentin's conjecture. If $G$ is a distance-regular graph with diameter $d$, then

$$c_2(G)^2=d^2 a\min_{j a\in\{1,\ldots,d\}}\left\{\frac{1-w_1(\theta_j)}{1-w_d(\theta_j)}\right\}.$$

The paper explicitly disproves this conjecture by giving counterexamples of diameter $4$ and larger, although it proves the equality for several families, including sufficiently large classical-parameter families.