Minimal-space characterization conjecture for perfect incremental samplers

Let $\mathbf{x} \in \mathbb{R}_+^n$ be updated by an incremental stream. Let $G$ be a nonnegative function on the coordinates, and write $G(\mathbf{x})=\sum_{u=1}^n G(\mathbf{x}(u))$. An $O(\log n)$-bit perfect $G$-sampler in the random oracle model samples an index $v\in[n]$ with probability

$$\frac{G(\mathbf{x}(v))}{G(\mathbf{x})}\pm\frac{1}{\operatorname{poly}(n)}.$$

Here $\mathcal{G}$ denotes the class of functions used for the paper's universal perfect samplers. Minimal-space characterization conjecture. If there is an $O(\log n)$-bit perfect $G$-sampler in the random oracle model, then $G\in\mathcal{G}$. This conjecture asks whether $\mathcal{G}$ captures all functions admitting minimal-size perfect samplers for incremental streams; the paper establishes universal perfect samplers for every function in $\mathcal{G}$, but does not establish the converse.