  Precise measurements of the branching ratios for the $K\to\pi\nu\bar{\nu}$
decays can provide unique constraints on CKM unitarity and, potentially,
evidence for new physics. It is important to measure both decay modes,
$K^+\to\pi^+\nu\bar{\nu}$ and $K_L\to\pi^0\nu\bar{\nu}$, since different new
physics models affect the rates for each channel differently. We are
investigating the feasibility of performing a measurement of
BR($K_L\to\pi^0\nu\bar{\nu}$) using a high-energy secondary neutral beam at the
CERN SPS in a successor experiment to NA62. The planned experiment would reuse
some of the NA62 infrastructure, including possibly the NA48 liquid-krypton
calorimeter. The mean momentum of $K_L$ mesons decaying in the fiducial volume
is 70 GeV; the decay products are boosted forward, so that less demanding
performance is required from the large-angle photon veto detectors. On the
other hand, the layout poses particular challenges for the design of the
small-angle vetoes, which must reject photons from $K_L$ decays escaping
through the beam pipe amidst an intense background from soft photons and
neutrons in the beam. We present some preliminary conclusions from our
feasibility studies, summarizing the design challenges faced and the
sensitivity obtainable for the measurement of BR($K_L\to\pi^0\nu\bar{\nu}$).
