Rationality of an Enriques–Fano threefold of genus five

We prove the rationality of a non-Gorenstein Fano threefold of Fano index one and degree eight having terminal cyclic quotient singularities and Picard group $\mathbb Z$. This threefold can be described as the quotient of a double covering of $\mathbb P^3$ ramified in a smooth quartic surface by an involution fixing eight different points.