Motivic Krasner formula for discriminant strata

Suppose that $k$ has characteristic $p>0$. Let $m\ge2$ be an integer, and let $\mathrm{Fie}_{m,d}\subset\mathrm{Fie}_{m}$ be the locus of degree-$m$ field extensions of $k((\pi))$ with discriminant exponent $d$.

Motivic version of Krasner's formula. In the Grothendieck-type ring $\mathcal{R}$,

$$[\mathrm{Fie}_{m,d}]=\begin{cases}1 & (p\nmid m,\ d=m-1),\\0 & (p\nmid m,\ d\ne m-1),\\(\mathbb{L}-1)\mathbb{L}^{\left\lfloor(d-m+1)/p\right\rfloor} & (p\mid m,\ p\nmid(d-m+1)),\\0 & (p\mid m,\ p\mid(d-m+1)).\end{cases}$$

This conjecture gives the motivic sizes of discriminant strata of local field extensions and is used for the paper's computations. The source gives no resolution.