Postnikov–Shapiro's graded Betti-number conjecture for power ideals

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic $0$ or sufficiently large. For a linear degree function $\phi$ satisfying

$$\phi(r)=l+k(n-r)>0,$$

let $I_\phi$ be the monomial ideal generated by $(x_{i_1}\cdots x_{i_r})^{\phi(r)}$ over all nonempty subsets $\{i_1,\ldots,i_r\}$ of $\{1,\ldots,n\}$, and let $J_\phi$ be generated by $(x_{i_1}+\cdots+x_{i_r})^{r\phi(r)}$ over the same subsets. Postnikov–Shapiro's graded Betti-number conjecture. The graded Betti numbers of $R/I_\phi$ and $R/J_\phi$ should agree. This conjecture is false in general: Schenck gave a counterexample, although the $n=3$ case leads to a more precise conjecture about the resolution of $J_\phi$.