Artinian fibre products with pure Betti cone: characterize Hilbert series

Determine whether, for an Artinian fibre product ring R = R1 ×_k R2 over a field k with R1 ≠ k and R2 ≠ k, the equality of the Betti cone of finitely generated graded R-modules with the pure Betti cone, namely B_Q(R) = B_Q^{pure}(R), necessarily implies that the Hilbert series of R is H_R(z) = 1 + n z for some integer n ≥ 0.

Background

The paper proves that for fibre products R with socle concentrated in degree 1 (i.e., (R) = 1), if the Betti cone equals the pure Betti cone, then R is Cohen–Macaulay with minimal multiplicity, specifically H_R(z) = (1 + n z) / (1 − z). Combined with earlier work (Kumar, 2017) showing that if (R) = 0 and B_Q(R) = B_Q^{pure}(R), then R is Artinian and level, the authors establish that fibre products satisfying B_Q(R) = B_Q^{pure}(R) are Cohen–Macaulay and level in the cases they treat.

Motivated by these results, the authors ask whether a stronger characterization holds in the Artinian fibre product case: namely, whether the equality B_Q(R) = B_Q^{pure}(R) forces the Hilbert series to be of the form 1 + n z, which would mirror a minimal multiplicity-type behavior for Artinian rings in this setting.