Conjecture on coefficients of multivalued-group polynomials

Let $p_n$ be the polynomial whose coefficients in the elementary symmetric functions are denoted by $A_{k_1,k_2,k_3}$, with $e_1$ the first elementary symmetric function. The integer $n$ is the degree parameter, and $p$ is a prime.

Coefficient divisibility and nonvanishing conjecture. 1. If $n=p^m$ is a power of a prime $p$, then every coefficient of $p_n$ except the coefficient of $e_1^n$ is divisible by $p$. 2. If $n$ is even, then all coefficients $A_{k_1,k_2,k_3}$ are nonzero.

These assertions describe arithmetic divisibility and nonvanishing patterns in the coefficients arising from the Newton-polyhedron analysis of multivalued groups. The supplied text does not state whether either assertion has been proved or refuted.