Boundary-intersection conjecture for affine flag fibers

Let $\fg$ be the Lie algebra under consideration, with index set $I$, and let $\fF^\lambda_{\fg,\fb}$ be the corresponding fiber space with open stratum $\overset{\circ}{\fF}{}^\lambda_{\fg,\fb}$. For a component $\bK$ of this open stratum, write $\ol{\bK}$ for its closure in $\fF^\lambda_{\fg,\fb}$, and let $\alpha_i$ denote the simple coroot associated with $i\in I$. Boundary-intersection conjecture. If $\lambda\neq 0$, then for every irreducible component $\bK$ of $\overset{\circ}{\fF}{}^\lambda_{\fg,\fb}$, there exists at least one $i\in I$ such that

$$\ol{\bK}\cap \overset{\circ}{\fF}{}^{\lambda-\alpha_i}_{\fg,\fb}\subset \fF^\lambda_{\fg,\fb}$$

is non-empty. The conjecture is needed to apply the uniqueness theorem cited in the source; no resolution is supplied in the provided text.