Excluding a line from gammoids

For all positive integers $\ell$ and $r$, we prove that a finite simple rank-$r$ gammoid with no $U_{2,\ell+2}$-minor has at most $\ell(r-1)+1$ elements. This gives an affirmative answer to Problem~7.2 of Boretsky and Walsh [European J. Combin.\ 137 (2026), 104419], for both gammoids and transversal