Asymptotic probability formula for conditioned convex hulls

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

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

Let ${\bf Q}^{\mathbf{K}}_{n,m}$ denote the probability that the convex hull of $n+m$ independent uniform points in $\mathbf{K}$ has $n$ vertices. Asymptotic probability conjecture. Under these hypotheses,

$$n^{-1}\left(\log\left({\bf Q}^{\mathbf{K}}_{n,\lfloor n\lambda\rfloor}\right)+2n\log(n)\right)\to\log\left(\frac{e^2}{4}\frac{(\lambda+1)^{\lambda+1}}{\lambda^\lambda}{\bf A}(C^\star)^\lambda{\sf L}(C^\star)^3\right).$$

This is presented as a conjectural asymptotic expansion for the conditioning probability; the subsequent convergence statement is explicitly made under its hypothesis.