  We describe the proximity effect in a short disordered metallic junction
between three superconducting leads. Andreev bound states in the multi-terminal
junction may cross the Fermi level. We reveal that for a quasi-continuous
metallic density of states, crossings at the Fermi level manifest as closing of
the proximity-induced gap. We calculate the local density of states for a wide
range of transport parameters using quantum circuit theory. The gap closes
inside an area of the space spanned by the superconducting phase differences.
We derive an approximate analytic expression for the boundary of the area and
compare it to the full numerical solution. The size of the area increases with
the transparency of the junction and is sensitive to asymmetry. The finite
density of states at zero energy is unaffected by electron-hole decoherence
present in the junction, although decoherence is important at higher energies.
Our predictions can be tested using tunneling transport spectroscopy. To
encourage experiments, we calculate the current-voltage characteristic in a
typical measurement setup. We show how the structure of the local density of
states can be mapped out from the measurement.
