The $2n$-conjecture over the rationals

Let $S$ be an $n\times n$ zero-nonzero pattern that is spectrally arbitrary with respect to $\mathbb{Q}$. The $2n$-conjecture over the rationals. Then $S$ has at least $2n$ nonzero elements. The analogous bound fails over $\mathbb{C}$ but remains open over $\mathbb{R}$; the rational case is presented as a conjecture believed to be within reach.