Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, other cubical groups.

HHG can be equipped with a boundary, defined there. This boundary is endowed with a topology. However, a HHG can be endowed with different hierarchically hyperbolic structures. The question I ask is the following:

Does the boundary depends on the HH structure, that is, given two choices of HH structures, are the boundaries homeomorphic ?

In the paper cited above where the boundary is defined, it is proved the following. CAT(0) cube complexes can be given a HH structure, using the so-called factor systems (see the first paper cited above). For different choices of factor systems, the boundary is the same (it is homeomorphic to the simplicial boundary, endowed with some topology). So, this is a partial answer to my question.

The reason I ask this is that I mainly speak relative hyperbolicity. Given a relatively hyperbolic group $$\Gamma$$, one can choose different relatively hyperbolic structures (this corresponds here to choosing different collections of peripheral subgroups). If the peripheral subgroups are amenable, then the Bowditch boundary does not depend on the relative hyperbolic structure, but in general it does.

• One nice important fact about the boundary of a hyperbolic group/space is that its metric is unique up to quasi-symmetric equivalence (in particular, its Hausdorff dimension is well-defined). So I'd not interpret the uniqueness of the boundary up to homeomorphism as a definite uniqueness result. – YCor Oct 11 '18 at 11:05
• I think it is open whether or not it depends on the HH structure. – Paul Plummer Oct 11 '18 at 15:12
• I think it is worthwhile to note that on page 4 of the paper itself you will find the following more structured version of your question: Given two HH structures on the same space $X$ (e.g. on the same group), does the identity map on $X$ extend continuously to a map from $X$ union one boundary to $X$ union the other, so that the map in turn restricts to a homeomorphism between the boundaries? – Lee Mosher Oct 12 '18 at 21:03
• It is open whether it depends on the HH structure. There are some cases where it's known to be independent of the HH structure (for example, if the space is hyperbolic, then the HH boundary wrt any HH structure is homeomorphic to the Gromov boundary). (@HJRW, probably I told you about the cubical case, e.g. in a RAAG/RACG it shouldn't matter whether you've used the naive HH structure one can obtain by coning off all parabolic subgroups, or the "minimal" HH cubical HH structure arising from the hyperplanes. But I don't remember either :-)) – Mark Hagen Oct 15 '18 at 18:52
• @MarkHagen Thank you very much for the clarification! I guess there is a lot to do then :) – M. Dus Oct 16 '18 at 18:23