Equality of the Borisov–Nuer and Enriques-image loci

Let $g=H_Y^2+1\geq 5$ be an odd integer, let $\mathcal F_g$ be the moduli space of polarized $K3$ surfaces of genus $g$, and let $\Sigma_g\subset\mathcal F_g$ be the naturally defined locus arising as the image of the Enriques-to-$K3$ map. Let $\Xi_g\subseteq\Sigma_g$ be the locus of polarized $K3$ surfaces $(X,H_X)$ for which there exist a fixed-point-free involution $\theta:X\to X$ and a line bundle $M\in\operatorname{Pic}(X)$ satisfying $\theta^*H_X\simeq H_X$, $\theta^*M\simeq M$, and $(X,H_X,M)\in\mathcal{BN}_g$. Equality conjecture. The two loci coincide:

$$\Xi_g=\Sigma_g.$$

The claim says that every point of the Enriques-to-$K3$ image arises from data satisfying the Borisov–Nuer condition, rather than merely giving the inclusion $\Xi_g\subseteq\Sigma_g$. The source does not specify a resolution of this conjecture.