The primitivity conjecture for extremal reciprocal polynomials of composite half-degree

Let $d$ be an even natural number, and let $P_d(x)$ be the minimal polynomial of an extremal reciprocal algebraic integer of degree $d$. A polynomial is primitive in the sense used by the source. Primitivity conjecture. If $d/2$ is composite and $d\notin\{8,12,18,20\}$, then $P_d(x)$ is not primitive.

The conjecture is motivated by the observation that $P_8(x)$, $P_{12}(x)$, $P_{18}(x)$ and $P_{20}(x)$ are the only primitive minimal polynomials observed for extremal reciprocals with composite half-degree. The source gives computational support but no proof.