Prime-degree extremal polynomial conjecture in the 5 modulo 6 case

Let $d\geq 17$ be prime with $d\equiv 5\pmod 6$. Define

$$P_d(x)=\frac{x^{d+2}-x^2-1}{x^2-x+1}.$$

Let an algebraic integer of degree $d$ that is not a root of unity be extremal if it has the minimum possible house among such algebraic integers. Prime-degree extremal polynomial conjecture. The extremal $\alpha$ has minimal polynomial $P_d(x)$.

The polynomial is established in the source to have a real root between $1$ and $\sqrt[d]{2}$, with its $d$th power tending to $2$; computations for $d=17$ and $23$ motivate the conjecture. No resolution status is given.