Existence of a minimizer for the local delta-invariant on a henselized klt pair

Determine whether the local \(\delta\)-invariant \(\delta(X,\Delta;\nu_0)\) admits a minimizing valuation when \(x\in(X,\Delta)\) is the henselization of a klt pair of a local scheme essentially of finite type over an algebraically closed field of characteristic zero and \(\nu_0\) is a nonzero valuation centered at \(x\).

Background

The main theorem establishes existence of a minimizer for log Fano cone singularities, while Example \ref{chatchat} shows that the analogous assertion can fail for a general local klt pair essentially of finite type. The paper then observes that, in the example, an étale cover does admit a minimizing valuation and asks whether passing instead to the henselization restores the minimizer-existence property.