Rationality conjecture for general Peskine sixfolds with trivial twisting

Let ${\mathcal D}^{1,6,10}$ be the parameter space appearing in the construction of Peskine sixfolds, and let $X_1^\sigma$ denote the Peskine sixfold associated to $\sigma$. For a general $\sigma\in{\mathcal D}^{1,6,10}$, let $(S,\beta)$ be its associated degree $6$ twisted K3 surface, where $\beta$ is the twisting class.

Rationality conjecture for Peskine sixfolds. If the twisting class $\beta$ is trivial, then $X_1^\sigma$ is rational.

This conjecture is motivated by the analogous expected relationship for cubic fourfolds containing a plane between rationality and triviality of the twisting class. The paper presents it as an open question for general Peskine sixfolds.