Extension of the upper bound for Hilbert coefficients of Gorenstein modules

Let $M$ be a finitely generated graded Gorenstein $R$-module, minimally generated by homogeneous elements of degree zero. Let $s=\operatorname{codim} M$ and $k=\left\lfloor\frac{s}{2}\right\rfloor$. Let $\beta_0(M)$ denote the minimal number of generators of $M$, and write $m=(t_i)_{i=0,\ldots,s}$. Let $\tilde t$ be the modified degree sequence used in the upper-bound theorem, and let $\Psi_{t}f_j(\tilde t)$ denote the corresponding bound for the $j$-th Hilbert coefficient. Extension conjecture. For all $0\leq j\leq d-1$, one has

$$e_j(M)\leq\frac{\beta_0(M)}{(s+1)!}\Psi_{t}f_j(\tilde t).$$

The preceding result establishes the analogous upper bound for the first Hilbert coefficient $e_1(M)$; the conjecture proposes its extension to all Hilbert coefficients in the stated range. The notation $\Psi_t f_j$ and the precise construction of $\tilde t$ are not defined in the supplied span, so the full bound requires checking against the paper.