Rings of coinvariants and $p$-subgroups

Let $\varrho :G\hookrightarrow GL(n, \mathbb{F})$ be a faithful representation of a finite group $G$ over the field $\mathbb{F}$ and let $V \cong \mathbb{F}^n$ be an $\mathbb{F}(G)$-module. It has been shown by L. Smith that if $n=3$ and the order of $G$ is divisible by the positive characteristic $p$ of $\mathbb{F}$, then $\mathbb{F} [V]^G$ is Cohen-Macaulay. Under the condition $n=3$ we prove the following conjecture through this remarkable result: If $\mathbb{F} [V]_G$ is a Poincaré duality algebra, then $\mathbb{F} [V]_{\operatorname{Syl}_{p}(G)}$ is a complete intersection, where $\operatorname{Syl}_{p}(G)$ is a Sylow $p$-subgroup of $G$.