**Question 1.** Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?

**Motivation.** I would like more generally to get an answer to the following question:

**Question 2.** Suppose $X$ is a smooth Fano variety with a $\mathbb C^*$-action. Let $Y\subset X$ be the connected component of $X^{\mathbb C^*}$ on which all the weights of the action are positive. Is it true that variety $Y$ can be deformed to a Fano variety?

A positive answer to the first question gives a negative to the second one.

**Explanation.** Indeed, a reflexive Delzant polytope by definition corresponds to a toric Fano. For any face of a Delzant polytope we can find a rational linear support function. This will give us a $\mathbb C^*$ action as in Question 2. A surface that deforms to a Fano surface has $b_2\le 9$. Finally, a toric surface corresponding to a polygon with $n$ sides has $b_2=n-2$.