The flatness conjecture for affine Weyl group S-cells

Let $W$ be the affine Weyl group, let $x\in W$, and let $[w,\mathbf{r}]\in\frak{P}$. Let $\calI(x)^+_{w,\mathbf{r}}$ be the corresponding space and let $\frak{c}_{w,\mathbf{r}}$ be its base. Flatness conjecture. For every $x\in W$ and $[w,\mathbf{r}]\in\frak{P}$, either

$$\calI(x)^+_{w,\mathbf{r}}=\emptyset,$$

or the projection

$$\calI(x)^+_{w,\mathbf{r}}\to\frak{c}_{w,\mathbf{r}}$$

is faithfully flat. The conjecture proposes a flatness property for the geometric spaces associated with S-cells in affine Weyl groups; its resolution is not indicated in the supplied text.