In "The universal Euler characteristic for varieties of characteristic zero", Bittner shows that over a field $k$ of characteristic zero, the Grothendieck ring $K_{0}(Var_{k})$ of varieties is generated by symbols
$[X] \in K_{0}(Var_{k})$,
where $X$ is smooth and proper, subject only to the relations that if $Y \subseteq X$ is a closed immersion of smooth proper varieties and $E \subseteq Bl_{Y}(X)$ is the exceptional divisor of the associated blow-up, then
$[X] - [Y] = [Bl_{Y}(X)] - [E]$.
Question: Is Bittner's result known to hold after inverting $p$ if $k$ is of positive characteristic $p$? That is, do we have an analogous description of $K_{0}(Var_{k}) \otimes_{\mathbb{Z}} \mathbb{Z}[1/p]$?