An *arc* is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is *planar* if it is contained in some plane.

The following questions are motivated by Anton Petrunin's https://mathoverflow.net/q/257420/36904 :

> **Question 1.** Does every topologically embedded sphere in
> $\mathbb{R}^3$ necessarily contain a ***planar*** arc?

 A negative answer to this question would immediately answer Anton's question in the negative as well.

> **Question 2.** Does every topologically embedded disk in
> $\mathbb{R}^3$ necessarily contain a ***planar*** arc?

**Remark.** Obviously, a positive answer to Question 2 would imply the same for Question 1, but the converse is not obvious, perhaps not even true.