Bounded-size conjecture for the set of minors of LT-superregular matrices

Let $\gamma\geq 2$ and let $p$ be an odd prime. For a vector $(a_1,a_2,\dots,a_{\gamma-1})\in\mathbb{F}_p^{\gamma-1}$, let $S_\gamma$ be the set associated with this vector in the paper, and let $N_\gamma$ be the corresponding number of relevant minors. Bounded-size conjecture. There exists a vector $(a_1,a_2,\dots,a_{\gamma-1})\in\mathbb{F}_p^{\gamma-1}$ such that $S_\gamma$ has at most $\frac{N_\gamma}{2}+2$ elements. This conjecture is one of two proposed from computational examples for small finite prime fields; its resolution is not given in the paper.