The Hilbert coefficient bounds conjecture

Let $N$ be a finitely generated graded Cohen–Macaulay $S$-module of codimension $s$ generated in degree $0$, with minimal and maximal shifts $m_i$ and $M_i$ in homological degree $i$, and let $h_i(d_1,\ldots,d_s)$ be defined by

$$h_i(d_1,\ldots,d_s)=\sum_{1\leq j_1\leq j_2\cdots\leq j_i\leq s}\prod_{k=1}^i(d_{j_k}-(j_k+k-1)).$$

Hilbert coefficient bounds conjecture. For $i=0,1,\ldots,n-s$,

$$\beta_0\frac{m_1m_2\cdots m_s}{(s+i)!}h_i(m_1,\ldots,m_s)\leq e_i(N)\leq\beta_0\frac{M_1M_2\cdots M_s}{(s+i)!}h_i(M_1,\ldots,M_s).$$

This is an analogue of the multiplicity conjecture for the higher Hilbert coefficients, extending the known formula for modules with pure resolutions. The source does not state whether these inequalities have been proved or disproved.