Excluding a line from gammoids

For all positive integers $\mathrm{\ell }$ and $r$, we prove that a finite simple rank-$r$ gammoid with no $U_{2,\mathrm{\ell }+2}$-minor has at most $\mathrm{\ell }(r-1)+1$ 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 matroids. The bound is sharp for every $\mathrm{\ell }$ and $r$. The key estimate is that, for every finite simple gammoid $M$, the sum of $\mathrm{\mid }L\mathrm{\mid }-2$ over its long lines is at most $\mathrm{\mid }E(M)\mathrm{\mid }-r(M)$.