Every infinitely edge-connected graph contains the Farey graph or $T_{{\mathrm{\aleph }}_0}\text{\hspace{0.17em}⁣}\ast t$ as a minor

We show that every infinitely edge-connected graph contains the Farey graph or $T_{{\mathrm{\aleph }}_0}\text{\hspace{0.17em}⁣}\ast t$ as a minor. These two graphs are unique with this property up to minor-equivalence.