For an $n$-variate polynomial $f = \sum_{a_1,\dotsc,a_n} x_1^{a_1}x_2^{a_2} \cdots x_n^{a_n}$, its Newton polytope $P_f$ is defined as the convex hull of all exponent vectors in the support of $f$. There are known examples where number of vertices of $P_f$ is low, while total number of monomials in $f$ is high. For example in Claim 4.4 of this paper, $|V(P_f)| = n$, while number of monomials are $n^{\Omega(\log n)}$. However, the example mentioned in this paper is not a symmetric polynomial. I wonder whether one can similarly find a symmetric polynomial with few vertices in its Newton polytope. More precisely,
Does there exist a symmetric polynomial $g$ with few vertices, say $|V(P_g)| = s$, while number of monomials in $g$ is at least $s^{\Omega(\log n)}$?
