Cell-structure conjecture for finite subset spaces of cell complexes

Let $X$ be a finite $n$-complex with $c$ components, each containing a single vertex. The finite subset space $\exp_k X$ is obtained from $\exp_{k-1} X$ by adding cells whose dimensions satisfy

$$k-c\leq i\leq nk.$$

Finite subset-space cell-structure conjecture. The space $\exp_k X$ has a cell structure obtained from $\exp_{k-1} X$ by adding cells of dimensions $k-c\leq i\leq nk$.

This conjecture describes the dimensions of the cells added at each stage of the finite subset-space filtration. The surrounding discussion states that the theorem on the connectivity of finite subset spaces follows from this conjecture together with Handel's inclusion result; its resolution is not specified in the source.