Arc-complexity conjecture for uniform matroids

Let $r,n\to\mathbb{N}$ with $n\geq r$. The uniform matroid of rank $r$ on $n$ elements is $U_{r,n}=({\{1,2,\ldots,n\}},\mathcal{I}_{r,n})$, where

$$\mathcal{I}_{r,n}=\{X\subseteq\{1,2,\ldots,n\}\mid |X|\leq r\}.$$

Let $T=\{1,2,\ldots,r\}$, $X=\{r+1,r+2,\ldots,n\}$, and $D=(X\cup T,X\times T)$. For a gammoid $M$, let $\mathrm{C}_A(M)$ denote its arc complexity. Arc-complexity conjecture for uniform matroids.

$$\mathrm{C}_A(U_{r,n})=r\cdot(n-r).$$

The displayed construction gives the upper bound $\mathrm{C}_A(U_{r,n})\leq r\cdot(n-r)$; the conjecture asserts that this representation is optimal. The authors state that they were unable to find a known graph- or digraph-theoretic result implying the matching lower bound.