Contractibility of sufficiently small Gromov–Hausdorff balls

Prove or disprove whether, for every compact metric space X, all sufficiently small open Gromov–Hausdorff balls centered at X are contractible.

Background

The paper proves that the Gromov–Hausdorff space is homeomorphic to Hilbert space and consequently has strongly contractible neighborhood bases. It explicitly notes that these neighborhoods need not be Gromov–Hausdorff balls.

The unresolved problem asks whether the balls themselves become contractible below a radius depending on the center, and whether they admit stronger contractions onto their centers.

References

For every $X\inM$, does there exist $r_0>0$ such that, for every $0<r<r_0$, the ball $(X,r)$ is contractible? More strongly, can each such ball be strongly deformation retracted onto ${X}$?

— The topology of Gromov--Hausdorff space  (2609.09639 - Ishiki, 9 Sep 2026) in Question 3, Section 6 (Questions)