For exchangeable sign-invariant random walks in higher dimensions, do the persistence probabilities obey a universal polynomial decay law whose exponent depends only on the dimension and the exchangeability structure?

Resolving this question would establish a higher-dimensional analogue of the Sparre Andersen universality theorem, one of the landmark results in fluctuation theory, and would significantly advance the understanding of first-passage phenomena for correlated, symmetric random processes. It would have downstream consequences for the analysis of multivariate storage systems, queueing networks, and reflected diffusions, connecting naturally to the obliquely reflected processes studied in the companion paper on BSVIs. It would also clarify which structural features of a stochastic process govern long-time positivity behavior, providing new invariance principles for a broad class of symmetric random systems.