Partitioning of plane sets into $6$ subsets of small diameter

In 1956, H. Lenz introduced a problem, which was to find members of the sequence
$$d_n=\inf_{\Phi}\{x\in \mathbb{R}^{+}:\Phi \subset \Phi_1 \cup \Phi_2 \cup \dots \cup \Phi_n, \forall i \,\textrm{diam}\, \Phi_i \leq x \}$$
where infimum is taken over all sets $\Phi$ of unit diameter. In this paper, we improve an upper bound for $d_6$ to $0.53432\dots$.