This question is about Joel Spencer's famous "six standard deviations" theorem. If you don't know the theorem, it's Theorem 1 in:

Spencer, Joel. Six standard deviations suffice. Trans. Amer. Math. Soc. 289 (1985), no. 2, 679–706. Full text PDF (open access)

As noted at the end of the paper, the constant K which Spencer obtained is actually K=5.32.

Question: does anybody know of a proof of the Theorem which gives a smaller value for the constant?