Weakening p-versality for infinite p-rank

Investigate whether the p-versality condition on a real-valued field extension can be weakened when the base field has infinite p-rank, and determine how such a weakening would characterize extensions that test geometric spectrality.

Background

The paper proves that a p-versal extension l/k can test geometric spectrality: for a weakly cartesian normed k-algebra A, the base change A_l is spectral if and only if A is geometrically spectral. The authors also observe that any extension with this testing property must at least be weakly p-versal, by considering normed algebras of the form k(a^{1/p}).

When k has finite p-rank, the paper states that weak p-versality and p-versality are equivalent, so the testing extensions are completely characterized in that setting. The unresolved issue concerns the infinite p-rank case, where the authors do not determine whether p-versality is necessary or whether a weaker condition suffices.