Conjecture on sensitivity to initial data for minimal solutions of the supercooled Stefan problem

Let $X_{0-}$ be a random variable with $\mathbb{E}|X_{0-}|<\infty$. For $x\in\mathbb{R}$, let $\underline{\Lambda}^x$ denote the minimal solution to the McKean–Vlasov equation with shifted initial condition $X_{0-}+x$.

Sensitivity conjecture. The map

$$x\mapsto \underline{\Lambda}^x$$

is continuous from $\mathbb{R}$ to $(D([-1,\infty)),\operatorname{M1})$.

This conjecture concerns sensitivity of the minimal solution to perturbations of the initial condition. The preceding result identifies the minimal solution as a physical solution when $\mathbb{E}|X_{0-}|<\infty$; continuity in the Skorokhod $\operatorname{M1}$ topology remains conjectural in the stated setting.