Existence of concrete realizations of full shifts

Determine whether, for every uniform hypergraph $S$, there exists a specialization of the generic longest-permutation matrix $(w_0)(v)$ over the ground field such that the resulting partial shift equals the full exterior shift, namely whether there always exists $v$ satisfying $\Delta_{(w_0)}(S)=\Delta_{(w_0)(v)}(S)$.

Background

The paper distinguishes the generic partial shift defined over a rational-function field from shifts obtained by specializing the indeterminates to elements of a chosen ground field. A field may be too small for any specialization to realize the generic shift for a given hypergraph and permutation; the example immediately demonstrates this phenomenon for a transposition over $GF(2)$.

The authors then note that they have not found an analogous obstruction for the full shift associated with the longest permutation $w_0$. This leaves unresolved whether every full exterior shift admits a concrete specialization of the longest-permutation matrix.