  We report the most precise measurements to date of the strong-phase
parameters between $D^0$ and $\bar{D}^0$ decays to $K^0_{S,L}\pi^+\pi^-$ using
a sample of 2.93 fb$^{-1}$ of $e^+e^-$ annihilation data collected at a
center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII
collider. Our results provide the key inputs for a binned model-independent
determination of the Cabibbo-Kobayashi-Maskawa angle $\gamma/\phi_3$ with $B$
decays. Using our results, the decay model sensitivity to the $\gamma/\phi_3$
measurement is expected to be between 0.7$^{\circ}$ and 1.2$^{\circ}$,
approximately a factor of three smaller than that achievable with previous
measurements. The improved precision of this work ensures that measurements of
$\gamma/\phi_3$ will not be limited by knowledge of strong phases for the next
decade. Furthermore, our results provide critical input for other
flavor-physics investigations, including charm mixing, other measurements of
$CP$ violation, and the measurement of strong-phase parameters for other
$D$-decay modes.
