Cellular structure conjecture for the infinite Apéry resolution

Let $S$ be a numerical semigroup, let $R$ be its numerical semigroup algebra, and consider the infinite Apéry resolution of the residue field over $R$. Cellular structure conjecture. One can ascribe a cellular structure to the infinite Apéry resolution. This would connect the combinatorial construction of the resolution with cellular resolutions from combinatorial commutative algebra, which resolve monomial ideals by supporting them on cell complexes. The source gives no further conditions or evidence establishing the conjecture.