The weight-one vector conjecture for inclusion matrices of arcs

Let $G$ be an arc, let $k$ and $p$ be as above with $p$ the prime such that $q$ is a power of $p$, and let $n$ be the nonnegative integer appearing in the inclusion matrix $\mathrm{M}_{n}$. Assume that

$$k \leqslant p+n(p-2)$$

and that $G$ has size $2k-3+n$.

Weight-one vector conjecture. The matrix $\mathrm{M}_{n}$ has a vector of weight one in its column space.

This conjecture is motivated by computational evidence for small $k$ and small $p$, and would extend known cases concerning the nonexistence of arcs of size $q+2$. Its general status is not specified in the source.