Toward Precision Measurement of $B_K$ with Quenched Kogut-Susskind Quarks

We present a status report of our ongoing effort toward a precision determination of the kaon $B$ parameter with the Kogut-Susskind quark action in quenched QCD. Results for $B_K$ so far accumulated at $\beta =5.85$, 5.93, 6.0 and 6.2 corresponding to $a^{-1}=1.3-2.6$ GeV do not exhibit the theoretically expected $O(a^2)$ behavior, but are apparently more consistent with an $O(a)$ behavior.