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).

**Edit.** Apparently the problem is still open. Here's an article (arXiv link), which contains a survey of some known results as of 2009. From the Introduction:

In 1966, Leo Moser asked for the region of smallest area which can accommodate
every planar arc of length one. The problem is known as “Moser’s worm problem” and is a variation of universal cover problems. In Moser’s problem, a cover is a set which contains a copy of any rectifiable planar arc of unit length, and is usually assumed to be convex. Such a minimal cover is known to have area between 0.2194 and 0.2738. However, the
original problem remains unsolved.