In his Ph.D thesis, A. Belshaw finds $K=5.199$, but he also claims there that Kai Uwe Schmidt was able to get $K$ as low as $3.65$. The idea of the process is described in Section 5.6 although the actual computations rely on "personal communications"communication".
See summit.sfu.ca/system/files/iritems1/13657/etd8131_ABelshaw.pdf