A cubical set $Box^{op} \to Set$ is a model for a homotopy type, via Grothendieck and Cisinski (here $Box$ is the box category with objects the natural numbers and arrows generated by face and degeneracy maps 'as usual'). A typical example is the singular cubical set of a space, $n \mapsto Hom(I^n,X)$. The homotopy groups of $X$ can be recovered from this cubical set as it satisfies a property analogous to that of Kan complexes (horns have fillers). In general, do the homotopy groups of a cubical set satisfying this 'Kan' condition agree with that of the homotopy type it represents?

$\newcommand\Box{\mathrm{Box}}\newcommand\Set{\mathrm{Set}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}$A cubical set $\Box\op \to \Set$ is a model for a homotopy type, via Grothendieck and Cisinski (here $\Box$ is the box category with objects the natural numbers and arrows generated by face and degeneracy maps ‘as usual’). A typical example is the singular cubical set of a space, $n \mapsto \Hom(I^n,X)$. The homotopy groups of $X$ can be recovered from this cubical set as it satisfies a property analogous to that of Kan complexes (horns have fillers). In general, do the homotopy groups of a cubical set satisfying this ‘Kan’ condition agree with that of the homotopy type it represents?