Necessity of valuative dimension in the negative K-theory vanishing bound

Determine whether there exists a scheme X with valuative dimension strictly larger than Krull dimension (i.e., vdim(X) > dim(X)) such that the negative K-group K_{-vdim(X)}(X) is nonzero; equivalently, ascertain whether the vanishing bound in Theorem B that uses valuative dimension is genuinely needed by constructing such an example or proving that none exists.

Background

The paper establishes a generalized Weibel vanishing result for negative K-groups: for a qcqs (derived) scheme X under mild topological assumptions, K_{-i}(X)=0 for all i > vdim(X), and X is K_{-vdim(X)}-regular. In the Noetherian case, vdim(X) coincides with dim(X), so the bound agrees with the classical conjectural expectation.

However, for non-Noetherian schemes, valuative dimension can exceed Krull dimension. The authors explicitly note that it is unclear whether this stronger bound using valuative dimension is necessary. Specifically, they do not know any example where vdim(X) > dim(X) and the negative K-group at degree -vdim(X) fails to vanish, which would demonstrate necessity of using the valuative dimension rather than the Krull dimension.