The extreme-point and convex-hull conjecture for laws of distinct values

Let $(X_1,X_2,\ldots)$ be an infinite exchangeable sequence, and let $K_n$ be the number of distinct values among its first $n$ terms. For $m\in\{1,2,\ldots,\infty\}$, let $K_{n,m}$ be the number of distinct values in $n$ independent samples from the uniform distribution on $m$ elements, let

$$\boldsymbol{v}_{n,m}=\bigl(\mathbb{P}(K_{n,m}=k):1\leq k\leq n\bigr),$$

and set $\boldsymbol{v}_{n,\infty}=(0,\ldots,0,1)$. Define

$$V_n=\{\boldsymbol{v}_{n,m}:m=1,2,\ldots,\infty\}$$

and let $\operatorname{conv}(V_n)$ denote its convex hull. Extreme-point and convex-hull conjecture. For $n\geq 3$, the set of extreme points of $\operatorname{conv}(V_n)$ is $V_n$, and the set of possible laws of $K_n$ for an infinite exchangeable sequence $(X_1,X_2,\ldots)$ is $\operatorname{conv}(V_n)$. This characterizes the possible distributions of the number of distinct values in a finite sample from an infinite exchangeable sequence; the supplied text presents it as the main open problem of the article, and gives no evidence of a resolution.