The resultant-square conjecture for the discriminant of a trigonal genus-three curve

Let $C$ be the trigonal genus-three curve defined by $f(x,y)=0$, with coefficients $\mu_1,\mu_2,\mu_3,\mu_4,\mu_5,\mu_6,\mu_8,\mu_9,\mu_{12}$. For variables $x$ and $y$, write $f_x$ and $f_y$ for the corresponding partial derivatives, and let $\operatorname{rslt}_z$ denote the resultant with respect to $z$. Define

$$\begin{aligned} R_1&=\operatorname{rslt}_x\big(\operatorname{rslt}_y(f(x,y),f_x(x,y)),\operatorname{rslt}_y(f(x,y),f_y(x,y))\big),\\ R_2&=\operatorname{rslt}_y\big(\operatorname{rslt}_x(f(x,y),f_x(x,y)),\operatorname{rslt}_x(f(x,y),f_y(x,y))\big),\\ R_3&=\gcd(R_1,R_2). \end{aligned}$$

Here $\operatorname{rslt}_z$ is the determinant of the Sylvester matrix with respect to $z$. Resultant-square conjecture. The polynomial $R_3$ has weight $144$ and is a perfect square in the ring

$$\mathbb{Z}[\mu_1,\mu_4,\mu_2,\mu_5,\mu_8,\mu_3,\mu_6,\mu_9,\mu_{12}].$$

The discriminant is a polynomial in the coefficients that vanishes exactly when the curve is singular, and the proposed construction is motivated by computer-algebra experiments in special cases. The source supplies no resolution status, so this remains open.