This question is about Joel Spencer's famous "six standard deviations" theorem. The theorem says that when $$ L_i(x_1,\dots,x_n) = a_{i1} x_1 + \dots + a_{in} x_n, \quad 1 \leq i \leq n, $$ are $n$ linear forms in $n$ variables with all $|a_{ij}| \leq 1$, then there exist numbers $\varepsilon_1,\dots,\varepsilon_n \in \{-1,+1\}$ such that $$ |L_i(\varepsilon_1,\dots,\varepsilon_n)| \leq K \sqrt{n} $$ for all $i$.

It is stated as 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?