Strictness of the stability inclusions for positive-dimensional arcs

For each $\mathfrak{x} \in \operatorname{\bf Arc}_{\kappa}$, the collections $StS_{\mathfrak{x}}$, $LStS_{\mathfrak{x}}$, and $WStS_{\mathfrak{x}}$ are defined by the corresponding stability conditions, with

$$\mathbbmss{SmSch}_{\kappa}\subset StS_{\mathfrak{x}} \subset LStS_{\mathfrak{x}} \subset WStS_{\mathfrak{x}} \subset \mathbbmss{Sch}_{\kappa}.$$

Strictness conjecture. If $\mathfrak{x}$ has positive dimension, each inclusion in this chain is strict. The smooth schemes form the basic class of $\mathfrak{x}$-stable schemes, and the conjecture predicts that each successive stability class is genuinely larger for positive-dimensional admissible arcs.