Expected Hilbert-series formula for relatively compressed algebras over general complete intersections

Let $R=k[x_1,\ldots,x_n]$, let ${\mathfrak c}\subset R$ be a general complete intersection of type $(d_1,\ldots,d_c)$, and let $A=R/I$ be relatively compressed with respect to ${\mathfrak c}$, with socle degrees $(s_1,\ldots,s_t)$. Put

$$e'=(n-c)s+d_1+\cdots+d_c-c.$$

Let $J\subset R$ be generated by general forms $G_1,\ldots,G_{c+t}$ of degrees $d_1,\ldots,d_c,e-s_1,\ldots,e-s_t$ and general forms $F_1,\ldots,F_{n-c}$ of degree $s+1$, assuming these forms form a minimal basis of $J$. The expected Hilbert-series conjecture. The Hilbert series of $A$ is

$$H_A(Z)=\left|\frac{\prod_{i=1}^c(1-Z^{d_i})(1-Z^{s+1})^{n-c}}{(1-Z)^n}\right|-\left|\frac{\prod_{i=1}^c(1-Z^{d_i})(1-Z^{s+1})^{n-c}\prod_{i=1}^t(1-Z^{e'-s_i})}{(1-Z)^n}\right|.$$

This is presented as an expectation in the general-complete-intersection case; the preceding bound follows from Fröberg's conjecture, so the equality remains open in the stated generality.