Motivic jet-space integral formula for formal power series

Let $k$ be a field of characteristic zero and let $f\in k[[x_1,\dots,x_d]]$ satisfy $f(O)=0$, where $O$ is the origin. For each $n\in\mathbb N^*$, let $f_n$ be the sum of the homogeneous parts of $f$ of degrees at most $n$, let $\mathscr X_{n,O}(f_n)$ be the variety of truncated arcs satisfying $f_n(\gamma)=t^n\bmod t^{n+1}$ and $\gamma(0)=O$, and let $\mathfrak X_f(n)$ denote the corresponding truncation of the formal scheme. Put

$$\omega=dx_1\wedge\cdots\wedge dx_d.$$

Motivic jet-space integral conjecture. The identity

$$[\mathscr X_{n,O}(f_n)]=\mathbb L^{(n+1)(d-1)}\int_{\mathfrak X_f(n)}\left|(\omega/df)(n)\right|$$

holds in $\mathscr M_k^{\mu_n}$. This would provide a common motivic integral description for the truncations associated with all polynomial approximations $f_n$, addressing the difficulty of proving rationality of the resulting series without a common log resolution.