Conjecture on the number of classes for curve orbits on a smooth Hermitian surface

Let $X_I$ be the smooth Hermitian surface and let $C_{F_J}$ denote the curve associated with $F_J$. Set $O={\rm Aut}(X_I)C_{F_J}$, and let $d$ be the number of nontrivial classes in the associated relations $\{R_i\}_{i=0}^d$.

Class-number conjecture. If

$$O={\rm Aut}(X_I)C_{F_J},$$

then

$$\nd=|\mathbb{P}^1(\mathbb{F}_{q^2})|\quad\text{for all }q,$$

that is, $(O,\{R_i\}_{i=0}^d)$ has $d=q^2+1$ classes.

The conjecture is motivated by the examples for $q=2$ and $q=3$, where the observed values of $d$ agree with $|\mathbb{P}^1(\mathbb{F}_{q^2})|=q^2+1$. Its validity for all $q$ is left open in the source.