Whereas I don't know of any recent progress in this problem, let me mention one result for 
*closed* curves.

> **Theorem.** A closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius  $L/4 - (\pi - 2)/2K$. 

This was proved in 1974 by H.H. Johnson ([link 1][1]) who used calculus of variations methods.  A geometric proof was given a bit later by Chakerian, Johnson and Vogt ([link 2][2]).

  


  [1]: http://www.ams.org/mathscinet-getitem?mr=0348631
  [2]: http://www.ams.org/journals/proc/1976-057-01/S0002-9939-1976-0402611-2/home.html%20