K-stability of Casagrande--Druel varieties

We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are n-dimensional varieties constructed from Fano double covers of dimension n−1. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of P^{n−1} ramified over smooth hypersurfaces of degree 2d with n>d>n2>1. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.