We study a certain two-parameter family of non-standard graded complete intersection algebras A(m, n). In case n = 2, we show that if m is even then A(m, 2) has the strong Lefschetz property and satisfies the complex Hodge–Riemann relations, while if m is odd then A(m, 2) satisfies these properties only up to a certain degree. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of A(m, n).