The reducible Specht modules for the Hecke algebra ${\mathcal{H}}_{\mathbb{C},-1}({\mathfrak{S}}_n)$

The reducible Specht modules for the Hecke algebra ${\mathcal{H}}_{\mathbb{F},q}({\mathfrak{S}}_n)$ have been classified except when $q=-1$. We prove one half of a conjecture which we believe classifies the reducible Specht modules when $q=-1$ and $\mathbb{F}$ has characteristic 0.