Weak Fröberg conjecture for Hilbert series of general ideals

Let $\mathcal{S}$ be the polynomial ring in $n+1$ variables, and let $f_1,\ldots,f_s$ be general forms of degrees $d_1,\ldots,d_s$. Set

$$I=(f_1,\ldots,f_s)\subseteq\mathcal{S}.$$

For a power series $P=\sum_{i\geq0}a_it^i$, define $\operatorname{coeff}_d(P)=a_d$; define $\lceil P\rceil$ by retaining coefficients until the first negative coefficient and replacing all subsequent coefficients by zero. Weak Fröberg conjecture. For every integer $d\geq0$,

$$\operatorname{coeff}_d\left(\mathrm{HS}_{\mathcal{S}/I}(t)\right)\geq\operatorname{coeff}_d\left(\left\lceil\frac{\prod_{i=1}^s(1-t^{d_i})}{(1-t)^{n+1}}\right\rceil\right).$$

This weaker inequality is considered as a separate version of Fröberg's conjecture. The source gives no resolution status beyond discussing it in the cited literature.