Stability of schemes smooth over fat points

Let $\mathfrak{n}$ be a fat point over $\kappa$, and let $X$ be a scheme smooth over $\mathfrak{n}$. Let $\mathfrak{x}$ be an admissible arc in $\operatorname{\bf Arc}_{\kappa}$ containing $\mathfrak{n}$. Stability conjecture. Every scheme $X$ which is smooth over a fat point $\mathfrak{n}$ is $\mathfrak{x}$-stable for every admissible arc $\mathfrak{x}$ containing $\mathfrak{n}$. Smooth schemes over a fat point are proposed as a broad source of stable schemes, extending the triviality phenomenon for local deformations of smooth schemes. The source explicitly presents this claim as conjectural, and no resolution is supplied here.