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We study a certain two-parameter family of non-standard graded complete intersections A (m, n). In case n=2, we show that A (m, 2) has the strong Lefschetz property and the complex Hodge-Riemann property if and only if m is even. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of A (m, n).
Abdallah et al. (Sun,) studied this question.