Kalai's separator conjecture for simple polytopes

Let $P$ be a simple $d$-polytope with $m$ vertices, and let $G_P$ be its graph. Kalai's separator conjecture. There exists a subset $V'$ of vertices of $P$ such that

$$|V'|=O\left(n^{1-\frac{1}{d-1}}\right)$$

and removing $V'$ from $G_P$ separates $G_P$ into two parts, each with at least $n/3$ vertices. The source presents this as a version attributed to Kalai and as a conjecture about vertex expansion; the variables $m$ and $n$ are inconsistent in the source statement, and no resolution is given.