Strongly robust toric ideals are the toric ideals for which the set of indispensable binomials is the Graver basis. The strongly robust simplicial complex $Δ_T$ of a simple toric ideal $I_T$ determines the strongly robust property for all toric ideals that have $I_T$ as their bouquet ideal. We prove that $\text{dim} Δ_T<\text{rank}(T)$ for configurations in general position, partially answering a question posed by Sullivant.