Lagrangian submanifolds in complex space forms satisfying an improved equality involving $\delta(2,2)$

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{5}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}\delta(2,2)\leq \text{\small${25}{4}$} H^{2}+8c,{align} where $H^{2}$ is the squared mean curvature and $\delta(2,2)$ is a $\delta$-invariant on $M$ introduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105-127]. The main purpose of this paper is to classify Lagrangian submanifolds of $\tilde M^{5}(4c)$ satisfying the equality case of the improved inequality (A).