Classification conjecture for symmetric positive period 1 polynomials

Let $P=P(x_1,\ldots,x_{n-1})$ be a symmetric polynomial with positive coefficients that generates a period $1$ seed. Symmetric-polynomial classification conjecture. The only possibilities are either

$$P=\sum_{i=1}^{n-1}x_i^2+M(x_1,\ldots,x_{n-1}),$$

where $M$ is any multilinear symmetric polynomial, or, when $n$ is odd,

$$P=\sum_{1\leq i<j\leq n-1}x_ix_j+A\sum_{i=1}^{n-1}x_i+B.$$

This proposes a classification within the symmetric positive case. The source gives no resolution.