The generic Hilbert series conjecture for ideals in the square-free algebra

Let $V_n$ be an $n$-dimensional vector space, let $\mathfrak{S}(V_n)$ denote the square-free algebra, and let $I_n$ be a generic ideal with generators of degrees $d_1,\dots,d_r$. Write $\langle F(t)\rangle$ for truncation before the first non-positive coefficient. Square-free Hilbert series conjecture. The Hilbert series of the quotient is

$$\frac{\mathfrak{S}(V_n)}{I_n}(t)=\left\langle(1+t)^n\prod_{i=1}^r(1-t^{d_i})\right\rangle.$$

The conjecture is motivated by replacing the square relations in a polynomial presentation of the square-free algebra by generic quadrics; the computations in the source support it, but the general assertion remains unproved there.