The Barratt-Milnor Sphere $X_n$ is an $n$-dimensional space which has non-vanishing singular homology in arbitrarily high dimensions. The space $X_n$ is a generalized Hawaiian Earring, i.e. the $n$-dimensional Barratt-Milnor Sphere $X_n$ is the one-point compactification of a countable collection of copies of $\mathbb{R}^n$. So $X_1$ is the classical Hawaiian Earring, for example.
Here, they prove that if $n > 1$ then the singular homology $H_q(X_n, \mathbb{Q})$ is non-zero whenever $q \equiv 1 \pmod{n-1}$. So $X_2$ has non-trivial singular homology groups for all $n \in \mathbb{N}$.
The following is the main question:
Is there a compact, connected metric space $X$ where some high-dimensional homology group doesn't vanish (higher than the dimension of the space), but where singular $\mathbb{Z}$-homology does eventually stabilize to zero?
If not, must the collection of $n$ such that $X$ has a non-trivial homology group at dimension $n$ contain an infinite arithmetic progression as the Barratt-Milnor Spheres do? That might be the key to a disproof, as it feels in the wheelhouse of some cohomological or higher-homotopy argument.
I was also wondering about these questions, which are of a more didactic nature:
Is there a more elementary proof that one of these spaces has non-vanishing homology in arbitrarily high dimensions? Their proof uses Whitehead products and coefficients from $\mathbb{Q}$.
What are other spaces where this occurs, especially where the 'reason it happens' is fundamentally different than for the Barratt-Milnor Spheres?
Is there a more complicated finite-dimensional space, but where the calculation that a higher-dimensional homology group doesn't vanish is legit-easy?
Can there be a one-dimensional space with such behavior? (EDIT: this was answered below in a comment)
Can we require the groups in such a space to be torsion (alternatively, torsion-free)?
A related question was asked here by someone else:
However, no reference was given for this part of the answer, and it's not clear to me why "all but finitely many homology groups" isn't just "all homology groups" or why homotopy-equivalent was used instead of "$n$-dimensional." Any insights?
EDIT: After two bounties went unanswered, I posted it on MO here. One of the experts in this sort of stuff responded that they think the factual questions are probably open rn. Sorry, didn't think that I was asking something that tricky! I figured such things would be well-known x)
https://mathoverflow.net/questions/383511/low-dimensional-spaces-with-high-dimensional-homology