Primitive-point conjecture for the unit sphere over finite fields

Let $q$ be a prime power with $q>25$, and let

$$x^2+y^2+z^2=1$$

be the unit sphere over ${\mathbb F}_q$. An ${\mathbb F}_q$-primitive point is a point on this sphere whose coordinates generate the extension required by the paper's definition of primitivity. Primitive-point conjecture. There exists an ${\mathbb F}_q$-primitive point on the unit sphere. The conjecture is motivated by computations, which found such points for all tested values of $q$ except $q=3,5,9,13,25$; the asserted existence for every prime power $q>25$ remains open.