Let $Y$ be a Gushel-Mukai threefold, we can either consider an ordinary Gushel-Mukai fourfold $X$ containing $Y$ as a hyperplane section, or we consider a special Gushel-Mukai fourfold $X'$ as double cover of a linear section of $\mathrm{Gr}(2,5)$ branched over $Y$. In both cases, the residue category of $X,X'$ are K3 category and the residue category of $Y$ is an Enriques category.

Consider another example, let $Z$ be a Verra threefold, its residue category is a Enriques category as Sasha explained in this post Semi-orthogonal decomposition of Verra threefold Now, consider a Verra fourfold $W$, which is a double cover of $\mathbb{P}^2\times\mathbb{P}^2$ with branch divisor $Z$. It is known that the residue category of $W$ is also a K3 category.

So my question is that does there exists other known pairs of Fano threefolds/fourfolds and the fourfold is related to the threefold such that the residue categories of one of them is K3 category and the other is a Enriques category?

For example, let's consider quartic double solid $V$, which is a double cover of $\mathbb{P}^3$ with branch divisor being a quartic K3 surface. The residue category of $V$ is an Enriques category. Is there a Fano fourfold associated to $V$ such that the residue category of it is a K3 category?