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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.

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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).