The generalized cylinder conjecture for prime fields and codimension-two parameters

Let $q$ be a prime power, let $r$ and $v$ be non-negative integers, and consider $q^r$-divisible spanning sets of $q^{r+1}$ points in $\operatorname{PG}(v-1,q)$. The generalized cylinder conjecture in the stated cases. The generalized cylinder conjecture is true for $(v,r,q)=(r+3,r,q)$ and for every $(v,r,p)$ when $p$ is prime. The paper presents structural reductions and proves the claim in these cases, while also giving counterexamples for suitable nonprime prime powers.