Per Alexandersson's MNability conjecture for Petrie functions times $p_2$

Let $G(k,m)\in\Lambda$ be the Petrie symmetric function, let $p_2$ be the second power-sum symmetric function, let $\operatorname{Par}$ denote the set of partitions, and let $s_\lambda$ denote the Schur function indexed by $\lambda$. Alexandersson's MNability conjecture. For every positive integer $k$ and every $n\in\mathbb{N}$, $G(k,m)\cdot p_2$ can be written as

\sumnonlimits\limits_{\lambda\in\operatorname{Par}}u_\lambda s_\lambda

with $u_\lambda\in\{-1,0,1\}$ for all $\lambda\in\operatorname{Par}$. The conjecture has been verified for all $k$ and $n$ satisfying $k+n\leq 30$, but the statement contains an apparent parameter mismatch, using $n$ in the quantifier and $m$ in $G(k,m)$; this should be checked against the source.