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.