According to Daniel R. Mauldin, Some Problems and Ideas of Erdős in Analysis and Geometry. Erdős Centennial 25 (2013): 365-376. this is still open. At least it was still open in 2013. In this chapter it is also stated that we can assume that $A=A_1\cup\cdots\cup A_n$ where the $A_i$ are compact convex sets, and that the statement is true if $n\leqslant 3$, while $n=4$ is the smallest open case.
