Coherent tensor structure on finite-height coefficient spaces

Construct a coherent tensor structure on the finite-height coefficient category of simplicial vector spaces [?]

Background

The paper studies the coefficient category $\mathbf{Vect}^{(n)}_k$ consisting of simplicial $k$-vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$. It observes that, for $n\ge 2$, the inherited levelwise tensor product does not preserve this finite homological-height condition: tensoring objects concentrated in degree $n-1$ can produce homology in degree $2n-2$. Consequently, the finite-height coefficient category does not inherit the ambient tensor product as a closed tensor structure.

The authors identify the construction of a separately defined coherent tensor structure on the finite-height coefficient category as a separate unresolved question. Such a structure would need to address the failure of closure under the levelwise tensor product while supplying the required higher coherence data.