The symplectic-invariant formula for the irregular spectral curve

Let $F_g(s,u,v)$ denote the genus-$g$ symplectic invariant of the spectral curve depending on parameters $s,u,v$, and let $\mathcal{M}_g$ be the moduli space of smooth genus-$g$ curves. Write $\chi(\mathcal{M}_g)$ for its Euler characteristic.

Symplectic-invariant conjecture.

$$F_g(s,u,v)=\chi(\mathcal{M}_g)\,s^{2g-2}\left(\left(1-\frac{u}{v}\right)^{2g-2}+\left(1-\frac{v}{u}\right)^{2g-2}-1\right).$$

The conjecture concerns the symmetric symplectic invariants of the regular spectral curve introduced in the surrounding discussion, and the source notes that it is true for $g=2$. No general proof or disproof is supplied.