Piecewise-triviality of truncation maps for affine smooth schemes

Let $X\in\mathbbmss{Sch}_{\kappa}$ be affine. Suppose that $X\to\mathfrak{n}$ and $X\to\mathfrak{m}$ are smooth morphisms, where $\mathfrak{n}\hookrightarrow\mathfrak{m}$ are elements of the point system $\mathbb{X}$ associated to an admissible arc $\mathfrak{x}\in\operatorname{\bf Arc}_{\kappa}$. Write $\ell(n)$ and $\ell(m)$ for the lengths of the fat points and let $d$ denote the relevant relative dimension. Piecewise-triviality conjecture. The natural morphism

$$\pi_{\mathfrak{n}}^{\mathfrak{m}}:\nabla_{\mathfrak{m}}X\to\nabla_{\mathfrak{n}}X$$

is a piecewise trivial fibration over $\kappa$ with general fiber

$$\mathbb{A}_{\kappa}^{d(\ell(m)-\ell(n))}.$$

This refines the earlier stability conjecture by describing the geometry of the truncation morphisms for affine schemes smooth over successive fat points. The supplied text gives no resolution of this refined conjecture.