Unitality via H-equivalence in the NP1 category

Determine whether every NP1-digital H-space (X, e, μ), where μ: X × X → X is NP1-continuous and μ ∘ (id_X, c_e) is NP1-homotopic to id_X and μ ∘ (c_e, id_X) is NP1-homotopic to id_X, is H-equivalent (via continuous pointed maps that are homotopy inverses and compatible with multiplication up to homotopy) to a unital NP1-digital H-space (Y, e_Y, μ_Y) satisfying μ_Y ∘ (id_Y, c_{e_Y}) = id_Y = μ_Y ∘ (c_{e_Y}, id_Y).

Background

The paper shows that any connected NP_i-digital H-space is H-equivalent to an irreducible H-space and further to a left-unital (or right-unital) irreducible H-space, but it does not establish simultaneous two-sided unitality. Using the classification in the NP2 setting, the authors prove that NP2-digital H-spaces are H-equivalent to unital H-spaces.

This leaves a gap for the NP1 setting: while left- or right-unital reductions are achievable, it remains explicitly open whether every NP1-digital H-space can be H-equivalent to a genuinely unital one (i.e., with both left and right unit identities holding exactly, not merely up to homotopy).