The LP optimum conjecture for odd-dimensional generators in $D_n$

Let $D_n$ be the classical association scheme whose elements include a set of generators $X$, and let $q$ be the underlying field parameter. For a positive integer $d$, a $d$-code is a subset of $X$ satisfying the distance condition used in the scheme, and let $\operatorname{LP}(d)$ denote the Delsarte linear-programming optimum for such codes.

LP optimum conjecture for $D_n$. If $n$ and $d$ are odd integers with $1\leq d\leq n$, then

$$\operatorname{LP}(d)=|X|\prod\limits_{i=1}^{\frac{d-1}{2}}\dfrac{q^{2i-1}-1}{q^{n+2i-1}-1}.$$

This conjecture asserts that the bound given in Corollary 3.4(d) of Schmidt, Weiss, and Steiner is sharp as the LP optimum for odd $n$. The formula was checked computationally for many small values of $q$, $n$, and $d$, but no general proof is supplied here.