Revised Vallentin conjecture for distance-regular graphs

Let $G$ be a distance-regular graph with diameter $d$, with eigenvalues $\theta_0>\theta_1>\cdots>\theta_d$ and cosine sequences $(w_0(\theta_j),\ldots,w_d(\theta_j))$ corresponding to the eigenvalues. Let $c_2(G)$ denote the least Euclidean distortion of the shortest-path metric of $G$. Revised Vallentin conjecture.

$$c_2(G)^2=\max_{r\in\{d-1,d\}}\left\{r^2\frac{1-w_1(\theta_1)}{1-w_r(\theta_1)}\right\}.$$

The maximum occurs at $r=d$ unless $G$ is antipodal. The paper proves this revised conjecture for diameter $3$ and gives partial results for diameter $4$.