Stabilization conjecture for dominant affine Bruhat intervals

Let $W=\mathbbm{Z}\Phi^\vee\rtimes W_f$ be an affine Weyl group, with coweight lattice $\Lambda^\vee$. Let $A_x$ be the alcove corresponding to $x$. For $\mu\in\Lambda^\vee$, define $x+\mu$ by $A_{x+\mu}=\mu+A_x$. An element $x\in W$ is dominant if $A_x$ lies in the dominant chamber, and a coweight is dominant if it lies in the closed dominant chamber. Write $[x,y]$ for a Bruhat interval.

Stabilization conjecture. For any dominant coweight $\lambda\in\Lambda^\vee$ and dominant $x,y\in W$, there exists an integer $N_0=N_0(x,y,\lambda)$ such that for every $N\geq N_0$,

$$[x+N_0\lambda,y+N_0\lambda]\cong[x+N\lambda,y+N\lambda].$$

The conjecture asserts eventual stabilization of translated dominant affine Bruhat intervals. The source presents it as a conjecture and supplies no resolution status in the given material.