Symmetric factorization conjecture for Gram determinants of planar curves

Let

$$R=\mathbb Z[d,x_1,x_2,y_1,y_2,z_1,z_2,z_3].$$

Let $R_1$ be the subgroup of elements invariant under $h_1,h_2,h_t,g_1,g_2,g_3$, and let $R_2$ be the subgroup of elements $w\in R$ satisfying

$$h_1(w)=h_2(w)=-w,\qquad h_t(w)=g_1(w)=g_2(w)=g_3(w).$$

Symmetric-factorization conjecture. The determinant has the following factorization properties: (1) $\det G_n=u^2-v^2$ for some $u\in R_1$ and $v\in R_2$; (2) it factors as

$$\det G_n=\prod_\alpha(u_\alpha^2-v_\alpha^2),$$

where $u_\alpha\in R_1$, $v_\alpha\in R_2$, and $u_\alpha-v_\alpha$ and $u_\alpha+v_\alpha$ are irreducible polynomials; and (3)

$$\det G_n=\prod_{i=1}^n(u_i^2-v_i^2)^{\binom{2n}{n-i}},$$

where $u_i\in R_1$ and $v_i\in R_2$. The source gives no resolution of these factorization assertions.