The inverse-polynomial Cheeger conjecture for zonotopes

Let $Z$ be a zonotope of the form

$$Z=\sum_{i=1}^k[-\bm{v_i},\bm{v_i}],$$

and let $G$ be its $1$-skeleton. Let $i(G)$ denote the Cheeger constant, or edge-expansion, of $G$. Inverse-polynomial Cheeger conjecture for zonotopes. The quantity $i(G)$ is at worst inverse polynomial in $k$; equivalently, there is a polynomial $p$ such that $i(G)\geqslant 1/p(k)$. The conjecture would provide the weak isoperimetric control needed for percolation and connectivity arguments on zonotopes. The source calls it a bold conjecture and gives no proof or resolution.