Unbounded arc-complexity conjecture for gammoids

Let $\mathrm{C}_A(M)$ denote the arc complexity of a gammoid $M=(E,\mathcal{I})$, where $|E|$ is its ground-set size. Unbounded arc-complexity conjecture for gammoids. For every $q\in\mathbb{Q}$ there is a gammoid $M=(E,\mathcal{I})$ such that

$$\mathrm{C}_A(M)\geq q\cdot |E|.$$

This conjecture asserts that no universal linear bound on arc complexity holds for all gammoids. Immediately before the statement, the paper asks whether, for each finite field $\mathbb{F}$, there is a constant $c_{\mathbb{F}}$ such that every gammoid representable over $\mathbb{F}$ satisfies $\mathrm{C}_A(M)\leq c_{\mathbb{F}}|E|$; the conjecture gives a contrary unboundedness assertion, but no resolution is supplied here.