Strong regularity conjecture for the graph of integral distances

Let $q$ be an odd prime power, let $\boldsymbol{\beta}$ be the quadratic form defining the integral-distance graph on $\mathbb{F}_q^m$, and let $\mathcal{A}(m,q)$ denote the corresponding common-neighbor count for nonisotropic pairs. If $\mathcal{B}(m,q)$ denotes the number of common neighbors of $0$ and an element $v$ with $\langle v,v\rangle=0$ in $\mathbb{F}_q^m\backslash\{0,v\}$, then for $m\ge 2$ we have

$$\mathcal{B}(m,q)=\begin{cases} \mathcal{A}(m,q)&\text{for }m\text{ even},\\ \mathcal{A}(m,q)-(-1)^{\frac{(q-1)(m-1)}{4}}\cdot q^{\frac{m-3}{2}}\cdot\frac{q^2-1}{4}& \text{for }m\text{ odd}. \end{cases}$$

Strong regularity conjecture. For even dimension $m$, the graph of integral distances $\mathfrak{G}_{m,q}$ is a strongly regular graph. The conjecture predicts that the common-neighbor counts needed for strong regularity agree in even dimension; the odd-dimensional correction term explains why the corresponding assertion is not made there.