The norm-form conjecture for the resultant polynomial $R_q$

Let $q$ be an odd prime, and let $R_q(y)$ be the resultant polynomial defined in the paper. Norm-form conjecture. There exist polynomials $h_1,h_2\in\mathbb Z[y]$ such that

$$R_q(y)=h_1(y^2)^2-qh_2(y^2)^2.$$

The polynomial $R_q$ is known to be even of degree $2q-2$, with explicitly determined leading and constant coefficients, and its reduction modulo $q$ is $y^{2q-2}$. The asserted integral norm-form representation is proposed from numerical evidence and remains open.