Let $G$ be a finite non-abelian group of order $n$. Given $g\in G$ we denote its order by $\mathrm{ord}(g)$.

Consider the group algebra $\mathbb{F}[G]$ for some field $\mathbb{F}$.

Given an ordering $g_1,\ldots,g_n$ of all the elements of $G$, consider the product $$ P=(1+g_1+g_1^2+\cdots+g_1^{\mathrm{ord}(g_1)-1})(1+g_2+\cdots+g_2^{\mathrm{ord}(g_2)-1})\cdots (1+g_n+\cdots+g_n^{\mathrm{ord}(g_n)-1}) \in \mathbb{F}[G].$$

Q: Let $p$ be a prime dividing $n$ and take $\mathbb{F}=\mathbb{F}_p$. Does there there exist an ordering $g_1,\ldots,g_n$ such that $P$ vanishes in $\mathbb{F}_p[G]$?

This question is motivated by an old question I asked.