The following question is question 9.8 from Miller's paper ``Some interesting problems '':

QuestionSuppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, D_{n+1} \subseteq D_n$, then does $\bigcap_{n}D_n$ have the fixed point property?

As it is stated in the paper, the problem dates from the 1920's and was discussed by Kuratowski, Mazurkiewicz, and Knaster.

I wonder to know the status the problem, and if any progress or partial results are obtained about it.

I also remember, when I was passing undergraduate topology course, our teacher stated a very general open problem about the fixed point property of some subsets of the plain, from which the above could follow easily. I do not remember it correctly, so it would be helpful if someone states it, and gives references related to it (the problem was roughly something like: If $X$ is a compact connected subset of the plane, whose complement is also connected, maybe with some extra assumptions on $X$, then $X$ has the fixed point property).