Continuity conjecture for the minimal loss function under additive perturbations

Let $\underline{\Lambda}(x)$ denote the minimal solution's loss function for the McKean--Vlasov problem with additive perturbation $x$, and let $M$ be the space of loss functions.

Continuity conjecture. The map

$$x \mapsto \underline{\Lambda}(x)$$

is continuous from $\mathbb{R}$ to $M$.

Such stability under additive perturbations of the initial condition would yield propagation of minimality without assuming uniqueness of the physical solution to the McKean--Vlasov problem. The source does not provide a resolution of this conjecture.