Chain conjecture for closures of cells in \mathbb{G}_T

Let $\mathbb{G}_T$ be the parameter space of graded quotients with Hilbert function $T$, and let $U_c$ and $U_e$ be cells in $\mathbb{G}_T$ of dimensions $c$ and $e$, respectively. Chain conjecture. The cells $U_c$ and $U_e$ satisfy $\overline{U_c}\supset U_e$ if and only if there is a chain of cells

$$\overline{U_c}\supset \overline{U_{c+1}}\supset \cdots \supset \overline{U_e}$$

such that $U_{i+1}$ has codimension $1$ in $\overline{U_i}$. This gives a proposed codimension-one characterization of the closure relation among cells; the supplied text does not state whether the conjecture has been proved or disproved.