Hausdorff convergence to the optimizing convex set

Let $\mathbf{K}$ be a compact convex set of area $1$, let $\lambda>0$, and suppose that $\operatorname{argmax}\Phi_{\lambda,\mathbf{K}}$ consists of the single element $C^\star$, where

$$\Phi_{\lambda,\mathbf{K}}(C)={\sf L}(C)^3{\bf A}(C)^\lambda.$$

Let $d_H$ denote the Hausdorff distance. Hausdorff limit-shape conjecture. Under the conditional law ${\sf Q}^{\mathbf{K}}_{n,\lfloor n\lambda\rfloor}$,

$$d_H\left({\sf CH}(U[n+\lfloor n\lambda\rfloor]),C^\star\right)\xrightarrow[n]{\text{in probability}}0.$$

This is the claimed geometric consequence under the hypothesis of the preceding asymptotic-probability conjecture; the source provides no independent resolution.