  We extend the original cylinder conjecture on point sets in affine
three-dimensional space to the more general framework of divisible linear codes
over $\mathbb{F}_q$ and their classification. Through a mix of linear
programming, combinatorial techniques and computer enumeration, we investigate
the structural properties of these codes. In this way, we can prove a reduction
theorem for a generalization of the cylinder conjecture, show some instances
where it does not hold and prove its validity for small values of $q$. In
particular, we correct a flawed proof for the original cylinder conjecture for
$q = 5$ and present the first proof for $q = 7$.
