Existence of connected NP1-digital H-spaces not H-equivalent to topological groups

Ascertain whether there exists a connected NP1-digital H-space that is not H-equivalent (via continuous pointed maps that are homotopy inverses and compatible with multiplication up to homotopy) to any NP1-digital topological group.

Background

The paper constructs many disconnected NP_i-digital H-spaces that are not H-equivalent to digital topological groups, and proves that in the NP2 category any connected H-space is contractible and hence H-equivalent to a one-point digital topological group.

For NP1, the authors do not know whether analogous connected counterexamples exist. This question targets the boundary between H-space structures and fully group-like structures under NP1 homotopy, asking whether connected NP1-digital H-spaces may fail to be H-equivalent to NP1-digital topological groups.