  Crucial inputs for a variety of $CP$-violation studies can be determined
through the analysis of pairs of quantum-entangled neutral $D$ mesons, which
are produced in the decay of the $\psi(3770)$ resonance. The relative
strong-phase parameters between $D^0$ and $\bar{D}^0$ in the decays
$D^0\rightarrow K^0_{S,L}\pi^+\pi^-$ are studied using 2.93~${\rm fb}^{-1}$ of
$e^+e^-$ annihilation data delivered by the BEPCII collider and collected by
the BESIII detector at a center-of-mass energy of 3.773 GeV. Results are
presented in regions of the phase space of the decay. These are the most
precise measurements to date of the strong-phase parameters in $D \to
K_{S,L}^0\pi^+\pi^-$ decays. Using these parameters, the associated uncertainty
on the Cabibbo-Kobayashi-Maskawa angle $\gamma/\phi_3$ is expected to be
between $0.7^\circ$ and $1.2^\circ$, for an analysis using the decay
$B^{\pm}\rightarrow DK^{\pm}$, $D\rightarrow K^0_S\pi^+\pi^-$, where $D$
represents a superposition of $D^0$ and $\bar{D^0}$ states. This is a factor of
three smaller than that achievable with previous measurements. Furthermore,
these results provide valuable input for charm-mixing studies, other
measurements of $CP$ violation, and the measurement of strong-phase parameters
for other $D$-decay modes.
