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) who used calculus of variations methods. A geometric proof was given a bit later by Chakerian, Johnson and Vogt (link 2).