2-faces of reflexive Delzant polytopes

Question

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 $b_2=n-2$.

Answer 1

Haase and Melnikov have proved here that every lattice polytope can be realized as a face of some reflexive polytope. They do this by an iterative procedure that increases the dimension by one until the polytope becomes reflexive. At each step the new polytope is a wedge over a facet of the previous one, and one can show that this operation takes Delzant polytopes to Delzant polytopes. So if you start with a smooth polygon with more than 11 edges will produce a positive answer to your first question.