Jöllenbeck's resolution conjecture for monomial rings

Let $S=k[x_1,\ldots,x_n]$, let $\mathfrak a=\langle m_1,\ldots,m_l\rangle$ be a monomial ideal, and set $A=S/\mathfrak a$. Let $R$ be the graded commutative algebra constructed from the Taylor resolution modulo the relations generated by $[Y_I,Y_J]$ whenever $\gcd(m_I,m_J)=1$. Let $F_\bullet$ be a multigraded minimal $A$-free resolution of $k$, with

$$F_i=\bigoplus_{\alpha\in\mathbb N^n}A(-\alpha)^{\beta_{i,\alpha}}.$$

For $J\subset\{1,\ldots,n\}$ with $|J|=l$, let $\alpha_J$ be its characteristic vector. Jöllenbeck's resolution conjecture. There is an isomorphism of $k$-vector spaces

$$F_i\operatorname{\cong}\bigoplus_{J\subset\{1,\ldots,n\}\atop |J|=l}\ \bigoplus_{u\in R\atop |u|=i-l}A\big(-\big(\alpha_J+\deg(u)\big)\big).$$

The conjecture proposes an explicit description of the multigraded minimal resolution of the residue field, and the paper states that it implies the Charalambous–Reeves Poincaré–Betti formula and a Hilbert-series formula. It is proved only for some classes of monomial algebras in the supplied context.