Endpoint minimum conjecture for cosine sequences of distance-regular graphs

Let $G$ be a distance-regular graph of diameter $d$ with eigenvalues $\theta_0>\theta_1>\cdots>\theta_d$. For the cosine sequence $(w_0(\theta_1),w_1(\theta_1),\ldots,w_d(\theta_1))$ of $\theta_1$, endpoint minimum conjecture.

$$\min_{1\leq r\leq d}\left\{\frac{1-w_r(\theta_1)}{r^2}\right\}=\min_{r\in\{d-1,d\}}\left\{\frac{1-w_r(\theta_1)}{r^2}\right\}.$$

The minimum occurs at $r=d$ unless $G$ is antipodal. This conjecture is proposed as one of two auxiliary conjectures supporting the revised Vallentin conjecture; no resolution is supplied in the given text.