Finite combinatorial dimension of nonpositively curved spaces

Determine which nonpositively curved spaces have finite combinatorial dimension.

Background

Combinatorial dimension is defined through the dimension of the injective hull and is central to the paper's fixed-point results for locally elliptic actions. The authors emphasize that, unlike virtual cohomological, geometric, or asymptotic dimension, combinatorial dimension has not been thoroughly studied in the context of groups.

A general characterization of nonpositively curved spaces with finite combinatorial dimension would clarify the scope of the paper's ellipticity theorem and identify additional classes of spaces to which the result could apply.

References

Which (nonpositively curved) spaces have finite combinatorial dimension?

— Locally elliptic actions, torsion groups, and nonpositively curved spaces  (2110.12431 - Haettel et al., 2021) in Section 1, Introduction, immediately after Corollary B

What is the combinatorial dimension of a median graph (i.e. the $1$-skeleton of a CAT(0) cube complex)?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 4, subsection “Geodesic bicombings”

What is the combinatorial dimension of Euclidean buildings?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 4, subsection “Geodesic bicombings”

Is there a local characterization of combinatorial dimension? More precisely, if $X$ is a contractible metric space which has locally combinatorial dimension bounded above by some $N \in N$, does $X$ has combinatorial dimension bounded above by $N$?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, final list of questions on combinatorial dimension, item 1

If $X$ is the $1$-skeleton of an $n$-dimensional CAT(0) cube complex, is the combinatorial dimension of $X$ equal to $2{n-1}$? What is the Helly hull of $X$?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, final list of questions on combinatorial dimension, item 2