Algebraic ellipticity of double covers of $(\mathbf P^1)^3$

We show that every smooth complex double cover of $(\mathbf P^1)^3$ branched along a smooth divisor of tridegree $(2,2,2)$ is algebraically Gromov elliptic. These irrational Fano threefolds form Mori–Mukai family 3-1 and have anticanonical genus seven and Picard rank three. Their three coordinate projections define degree-four del Pezzo fibrations. Each threefold also admits a dominant rational map of degree two from $\mathbf P^3$.