Disk covering problem $r(6)$

The smallest value $r(6)$ such that $6$ disks of radius $r(6)$ can be arranged to cover the unit disk.

Updates

• Upper bound: ${\approx}0.555905$
  Reference unknown
  [via Circles Covering Circles, Erich Friedman]
• 1979 Lower bound: ${\approx}0.555905$
  Károly Bezdek in 1979
  [via Circles Covering Circles, Erich Friedman]