Unisingular Specht Modules

Let $G$ be a finite group and $\rho:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $\rho$ is unisingular if $\det(1-\rho(g)) = 0$ for all $g \in G$. Building on previous work in \cite{cullinan}, we consider the symmetric groups $S_n$ and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.