Primitive-matrix generation conjecture for integral vectors

Let $v$ be an $n$-dimensional integral vector whose entries are all nonzero modulo $p$. Suppose that

$$v^{\mathsf T}e\equiv 0\pmod{p},\qquad v^{\mathsf T}v\equiv 0\pmod{p}.$$

Here $e$ denotes the all-one vector, and a vector generates a primitive matrix when it occurs as the relevant generating vector for a primitive matrix in the paper's construction. Primitive-matrix generation conjecture. The following hold: (i) if $n\leq 8$, then $v$ can always generate some primitive matrix; and (ii) if $n\geq 2p+1$, then $v$ cannot generate any primitive matrix. The conjecture is motivated by computational experiments on graphs in the family studied in the paper; the stated criteria are proposed for further study and are not proved in the source.