On two approaches to classification of higher local fields

This article links Kurihara's classification of complete discrete valuation fields and Epp's theory of elimination of wild ramification.
For any complete discrete valuation field $K$ with arbitrary residue field of prime characteristic one can define a certain numerical invariant $\Gamma(K)$ which underlies Kurihara's classification of such fields into $2$ types: the field $K$ is of Type I if and only if $\Gamma(K)$ is positive. The value of this invariant indicates how distant is the given field from a standard one, i.e., from a field which is unramified over its constant subfield $k$ which is the maximal subfield with perfect residue field. (Standard $2$-dimensional local fields are exactly fields of the form $k\{\{t\}\}$.)
We prove (under some mild restriction on $K$) that for a Type I mixed characteristic $2$-dimensional local field $K$ there exists an estimate from below for $[l:k]$ where $l/k$ is an extension such that $lK$ is a standard field (existing due to Epp's theory); the logarithm of this degree can be estimated linearly in terms of $\Gamma(K)$ with the coefficient depending only on $e_{K/k}$.