The finite-hyperplane approximation conjecture for graph-statistics limit objects

Let $\mathbf{F}$ be a vector of graphs, and let $P_{\mathbf{F};\infty}$ be its limit object, a compact convex set whose volume is measured in its ambient finite-dimensional space. Finite-hyperplane approximation conjecture. There is a positive integer $m$ such that, for every $\varepsilon>0$, the limit object $P_{\mathbf{F};\infty}$ can be chopped down by $m$ hyperplanes to a polytope with finitely many vertices, removing volume at most $\varepsilon$. The conjecture proposes that arbitrary volume accuracy can be achieved with a number of cutting hyperplanes depending only on $\mathbf{F}$, not on $\varepsilon$; the source does not provide a resolution.