Nonexistence conjecture for Ehrhart-polynomial orthogonal polynomial sequences

Let $(P_j)_{j=0}^{\infty}$ be a family of reflexive polytopes whose Ehrhart polynomials $f_j(x)=H_{P_j}(x)$ form an orthogonal polynomial sequence and satisfy

$$f_j=M_j(2x+1)f_{j-1}+(1-M_j)f_{j-2},\qquad j=2,3,\dots,$$

with $M_j\in\mathbb{Q}$. Nonexistence conjecture. There do not exist such families for which $M_2=\frac{3}{8}$ or $M_2=\frac{7}{8}$. The possible values arise from the classification of two-dimensional reflexive polytopes; the source notes that other values, including $\frac{4}{8},\frac{5}{8},\frac{6}{8}$ and $\frac{8}{8}$, do occur.