Square-root recurrence conjecture for extremal reciprocal algebraic integers

Let $\mathrm{mr}(d)$ be the minimum of the houses of reciprocal algebraic integers of degree $d$ that are not roots of unity. An algebraic integer attaining $\mathrm{mr}(d)$ is called extremal reciprocal. If its minimal polynomial is denoted by $R_d(x)$, then, for even $d\geq 8$, the square-root recurrence conjecture states that an extremal reciprocal $\alpha$ of degree $d$ gives an extremal reciprocal $\sqrt{\alpha}$ of degree $2d$, with

$$R_{2d}(x)=R_d(x^2).$$

The claim is suggested by computations through even degree $34$ and describes a recurrence for the extremal reciprocal polynomials. No resolution status is given in the source.