  We establish existence and uniqueness of the solution to the cavity equation
for the random assignment problem in pseudo-dimension $d>1$, as conjectured by
Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and W\"astlund
(Annals of Mathematics, 2012). This fills the last remaining gap in the proof
of the original M\'ezard-Parisi prediction for this problem (Journal de
Physique Lettres, 1985).
