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?

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