Truly linear sparsifier conjecture for set systems

Let $\mathcal S \subseteq 2^{[m]}$ be a set system, let $w:[m]\rightarrow\mathbb{R}_{\geq 0}$ be weights, and let $\epsilon>0$. Write $\operatorname{CL}(\mathcal S)$ for the chain length of $\mathcal S$. A $(1\pm\epsilon)$ sparsifier retains a selected subset of coordinates while approximating the relevant set-system quantities within multiplicative error $1\pm\epsilon$. Truly linear sparsifier conjecture. There exists a $(1\pm\epsilon)$ sparsifier of $\mathcal S$ retaining only

$$O\left(\frac{\operatorname{CL}(\mathcal S)}{\epsilon^2}\right)$$

many coordinates. This conjecture seeks an analogue for arbitrary set systems of the $O(n/\epsilon^2)$ edge bounds for graph cut sparsification; the paper's results give sparsifiers whose size depends on accuracy and chain length, but do not establish this bound in general.