On the dimension of the strongly robust complex for configurations in general position

Strongly robust toric ideals are the toric ideals for which the set of indispensable binomials is the Graver basis. The strongly robust simplicial complex ${\mathrm{\Delta }}_T$ 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 $\text{dim}{\mathrm{\Delta }}_T<\text{rank}(T)$ for configurations in general position, partially answering a question posed by Sullivant.