The octanomial exclusion conjecture for extremal reciprocal algebraic integers

Let an extremal reciprocal primitive be an extremal reciprocal algebraic integer that is primitive, and let an octanomial be a polynomial with eight nonzero monomials. Octanomial exclusion conjecture. An extremal reciprocal primitive of degree $d$ cannot be a root of either a reciprocal octanomial of degree $D_1<2d$ whose inner monomials all have minus signs, or a reciprocal octanomial of degree $D_2<1.5d$ whose monomials all have plus signs.

The source presents this as a conjectural pattern based on computations, while a proposition later proves a version only under additional bounds on $d$, $D_1$ and $D_2$.