Adaptive-radius homology computation conjecture for intersections of balls

Let $\mathcal{X} \subset R^{n+1}$ be a finite set, let $A>0$, and let $\epsilon(x)$ be a positive $A$-Lipschitz continuous function. For each $x\bin\mathcal{X}$, write $B(x,\epsilon(x))$ for the corresponding ball.

Adaptive-radius homology computation conjecture. One can compute the homology of

$$\cap_{x\in \mathcal{X}}B(x,\epsilon(x))$$

with $\mathcal{X}^{O(n)}$ operations.

This conjecture proposes that the complexity bound known for unions of equal-radius balls extends to intersections of balls with radii varying according to a positive Lipschitz function. The source gives no resolution or further evidence for the conjecture.