The birational compatibility conjecture for Enriques moduli maps

Let $\mathcal{M}_{{\text{En}}}(2)^{{\text{ndeg}}}$ be the moduli space of non-degenerate degree-two Enriques surfaces, let $\mathcal{X}_4$ be the moduli space of pointed genus-$4$ curves, and let $\Psi_1$, $\Psi_2$, and $\Phi_1$ be the rational maps described above. Let $\Upsilon$ be the birational isomorphism from the displayed birational identification. Birational compatibility conjecture.

$$\Upsilon\circ\Phi_1=\Psi_2\circ\Psi_1.$$

The proposed equality identifies the composite construction through pointed genus-$4$ curves with the map obtained from the Enriques moduli space. The excerpt motivates it by comparing the degrees, but gives no evidence that it has been proved or disproved.