The decay mode $D^0 \to K_{\rm L}^0\pi^+\pi^-$ plays a crucial role in precision measurement of the CKM angle $\gamma$, which measures CP violation in weak quark interactions, using $B \to DK$ decays. The strong-phase parameters related to $D$ decays are inputs to the $\gamma$ measurement, and are measured in a model-independent approach. In this talk, I will show the role of a $D^0 \to K_{\rm L}^0\pi^+\pi^-$ amplitude model in these model-independent strong-phase measurements. A study of the resonant amplitude structure of the decay mode $D^0 \to K_{\rm L}^0\pi^+\pi^-$, using quantum-correlated $D^0\bar{D}^0$ data produced at the $\psi(3770)$ resonance ($\sqrt{s}=3.773$ GeV) is presented. The data sample was collected by the BESIII experiment and corresponds to an integrated luminosity of $2.93$~fb$^{-1}$. This study is the first amplitude analysis of a decay mode involving a $K_{\rm L}^0$, which also results in the first measurement of the complex {\it U-spin breaking parameters} ($\hat{\rho}$) related to various CP-eigenstate resonant modes through which the three-body decay proceeds. Contrary to the previou sly assumed values of unity, the moduli of the $\hat{\rho}$ parameters have central values in a wide range from 0.4 to 12.1, which indicates substantial U-spin symmetry breaking. Fractional resonant contributions and average strong-phase parameters over regions of phase space for both $K_{\rm S}^0\pi^+\pi^-$ and $K_{\rm L}^0\pi^+\pi^-$ modes are reported. Ratio of the branching fractions between $K_{\rm S}^0\pi^+\pi^-$ and $K_{\rm L}^0\pi^+\pi^-$ decay modes and the CP-even fraction of the $K_{\rm L}^0\pi^+\pi^-$ state calculated using the U-spin breaking parameters are also presented.