Dual F-signature analogue of the reduction-mod-p implication

Determine whether the reduction-mod-p implication established for F-signature under pure maps and klt-type hypotheses also holds for dual F-signature, specifically whether positivity of the dual F-signature of the reductions of the source rings forces the corresponding positivity conclusion for the reductions of the target rings in the non-module-finite setting.

Background

The paper proves a reduction-mod-p result for dual F-signature in a module-finite setting: when a regular ring is finite over a Cohen–Macaulay ring, the dual F-signature of suitable characteristic-p reductions is expressed through an asymptotic surjective number, and normality yields a positive lower bound.

The authors then note that the corresponding F-signature statement has been strengthened by a result allowing the map from the target to the source to be pure and the source to be of klt type, without requiring module-finiteness. They explicitly state that it is unknown whether an analogous statement holds for dual F-signature.