Nonexistence conjecture for distance-regular graphs with specified classical parameters

Let $q$ and $D$ be integers with $q \ge 2$ and $D \ge 4$. A distance-regular graph with classical parameters

$$\Big(D, q, q, \frac{q^2 (q^D-1)}{q-1} \Big)$$

is a graph whose classical parameters are given by this quadruple. Nonexistence conjecture. There exists no distance-regular graph with classical parameters

$$\Big(D, q, q, \frac{q^2 (q^D-1)}{q-1} \Big).$$

The conjecture is motivated by computational evidence that the corresponding valency and eigenvalue multiplicity are never integers for $q,D\le 3000$; the theorem proved in the paper shows that, under the paper's hypotheses, this is the only remaining parameter family when $D\equiv 0\pmod 6$ and the auxiliary parameter $\alpha$ is nonzero.