Cheltsov's nonrationality conjecture for Fano-Enriques threefolds

Let $X$ be a Fano-Enriques threefold whose canonical covering $V$ is smooth. Suppose that $V$ is one of the following varieties: the double covering of a quadric ramified in a divisor of degree $8$; the complete intersection of three quadrics in $\mathbb{P}^6$; or the double covering of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ ramified in a divisor of degree $(2,2,2)$.

Cheltsov's conjecture. In each of these three cases, $X$ is not rational.

This conjecture concerns the remaining smooth-covering cases after known rationality and nonrationality results for Fano-Enriques threefolds. The source attributes it to I. Cheltsov; its resolution status is not specified in the supplied text.