Degree formulas for the Euclidean joint signature of a generic plane curve

Let $\mathcal{J}^{\mathcal{E}}_d$ denote the Euclidean joint signature for a generic plane curve of degree $d$. The multiprojective witness-set dimensions are $\boldsymbol{e}_1=(1,1,1,1,0,0)$ and $\boldsymbol{e}_2=(0,1,1,1,1,0)$. Euclidean joint-signature degree conjecture. For $d\ge 3$,

$$\deg \overline{\mathcal{J}^{\mathcal{E}}_d}=12d(d^3-1),$$ $$\deg_{\boldsymbol{e}_1}\mathcal{J}^{\mathcal{E}}_d=8d^2(d^2-1),$$ $$\deg_{\boldsymbol{e}_2}\mathcal{J}^{\mathcal{E}}_d=4d(d-1)(3d^2+d-1).$$

These formulas were suggested by monodromy computations, which were successfully run for curves of degree up to $10$; their validity for all $d\geq 3$ remains open.