Strong 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. Strong Fröberg conjecture. For every integer $d\geq0$,

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

The formula is known for complete intersections and several additional cases, including two and three variables and the first relevant degree, but remains open in general. The source attributes the conjecture to Fröberg (1985).