Rationality conjecture for Gushel fourfolds with associated K3 surfaces

Let $\mathcal M_4$ be the moduli space of Gushel fourfolds, and let $(\mathcal M_4)_d$ denote the locus of discriminant $d$; the loci with superscripts $'$ and $”$ are the corresponding components appearing in the statement. An associated K3 surface is a K3 surface associated to the fourfold in the sense of the paper.

Rationality conjecture. A fourfold $[X]\in\mathcal M_4$ is rational if and only if it has an associated K3 surface, equivalently if and only if $[X]$ belongs to

$$(\mathcal M_4)_{10}^{'} \cup (\mathcal M_4)_{10}^{”} \cup (\mathcal M_4)_{20} \cup (\mathcal M_4)_{26}^{'} \cup (\mathcal M_4)_{26}^{”} \cup (\mathcal M_4)_{34}^{'} \cup (\mathcal M_4)_{34}^{”} \cup \cdots.$$

This is an analogue for Gushel fourfolds of the Kuznetsov conjecture for cubic fourfolds, relating rationality to the existence of an associated K3 surface. The source presents the assertion as a conjecture; its resolution is not established by the supplied text.