Algebraic ellipticity of double covers of ( P¹)³

We show that every smooth complex double cover of ( P¹)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 P³.