Lagrangian warped product immersions in ${\mathbb{S}}^6$

We study Lagrangian immersions in the nearly K"ahler ${\mathbb{S}}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature $\frac{1}{16}$ or they satisfy equality in Chen’s inequality, in which case the immersion is given explicitly.